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Research ArticleRobustness of Supercavitating Vehicles Based onMultistability Analysis

Yipin Lv Tianhong Xiong Wenjun Yi and Jun Guan

National Key Laboratory of Transient Physics Nanjing University of Science and Technology Nanjing 210094 China

Correspondence should be addressed to Tianhong Xiong xiongtianhongnjusteducn

Received 23 January 2017 Accepted 9 March 2017 Published 5 April 2017

Academic Editor Nikos Mastorakis

Copyright copy 2017 Yipin Lv et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Supercavity can increase speed of underwater vehicles greatly However external interferences always lead to instability ofvehicles This paper focuses on robustness of supercavitating vehicles Based on a 4-dimensional dynamic model the existenceof multistability is verified in supercavitating system through simulation and the robustness of vehicles varying with parametersis analyzed by basins of attraction Results of the research disclose that the supercavitating system has three stable states in someregions of parameters space namely stable periodic and chaotic states while in other regions it has various multistability suchas coexistence of two types of stable equilibrium points coexistence of a limit cycle with a chaotic attractor and coexistence of1-periodic cycle with 2-periodic cycle Provided that cavitation number varies within a small range with increase of the feedbackcontrol gain of fin deflection angle size of basin of attraction becomes smaller and robustness of the system becomes weaker Inpractical application robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initiallaunching conditions

1 Introduction

When vehicle navigates underwater at a high speed waterpressure on surface of supercavitating vehicle will decreaseOnce the speed is increased to a critical value pressure ofwater will reach the level of vaporization then the water willchange from liquid phase to vapor phase which is calledcavitation [1] With unceasing increase of speed as shown inFigure 1 the cavity will move backward and expand alongthe surface of vehicle and finally develops into supercavity[2 3] which envelops the whole vehicle Different conditionsof flow field and geometric shapes of vehicles lead to varioustypical states of supercavity almost completely In study ofcavitating flow a dimensionless parameter 120590 is often usedto characterize the extent of cavitation namely cavitationnumber defined as120590 = 2(119901infinminus119901119888)1205881198812 where120588 is the densityof water 119901infin is the ambient pressure at infinity 119901119888 is the cavitypressure and 119881 refers to the velocity of vehicle [4] Afterformation of supercavity the resistance of water turns intothe resistance of cavity consisting of vapor which increasesthe velocity greatly and enables supercavitating vehicle to

ldquoflyrdquo underwater Despite remarkable reduction of resistanceby supercavity supercavitating vehicle in high speed will becovered by supercavity substantially or wholly which resultsin decrease of wetting area loss of most buoyancy andforward drift of pressure center decrease of attached qualityand damping torque accordingly All these factors makesupercavitating vehicle sensitive to ambient interferencesrobustness of supercavitating vehicle is thus weakened [5 6]In addition change in fin deflection angle always affectsattitude of vehicle inside supercavity Change in cavitationnumber also affects the size of supercavity which leads tocollision between tail of vehicle and supercavity and results inthe generation of nonlinear planning force Nonlinear plan-ning force brings not only greater friction to supercavitatingvehicle but also vibration and shock as well as complex non-linear physical phenomena such as bifurcation and chaoswhich poses a challenge to dynamic modeling guidanceand control [7 8] Therefore efficient control on attitude ofsupercavitating vehicle inside supercavity improvement ofrobustness of motion and reduction of shock from collisionbetween supercavitating vehicle and supercavity are critical

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 6894041 13 pageshttpsdoiorg10115520176894041

2 Advances in Mathematical Physics

Figure 1 Illustration of vehicle inside supercavity

to ensure the stable motion of underwater supercavitatingvehicle

Unique navigation environment of supercavitating vehi-cle has predetermined that it is a multivariable system withcomplex couplings and uncertain hydrodynamic parametersSupercavitating vehicle is different fromcommonunderwatervehicle both particularities of control object such as nonlin-earity of the planning force anduncertainty of supercavitatingvehicle as well as the robustness with external interferencesshould be taken into consideration There are a few paperspublished regarding the study of robustness Linear state-feedback controlmethod presented by Lin et al [9] achievedstrong robustness in error of modeling for planning forceof supercavitating vehicle Vanek and Balas [10] considereduncertainty in shape of supercavity and the control withlinear parameter variables (ldquoLPVrdquo) after linearization offeedback was researched Goel [11] applied robustness controlto the design and analysis of linear model of supercavitatingvehicle with target to uncertainty of its hydrodynamic param-eters Zhao et al [12] considered the uncertainty (existingin linear model of supercavitating vehicle) and unmodeleddynamics (mainly displayed by hydrodynamic coefficientperturbation in simulation) and applied robustness control tothe design of controllerWang andZhao [13] applied principleof minor perturbation to linearization of longitudinal motionmodel of supercavitating vehicle and control methods werefurther researched based on linear feedback theory androbustness control theory Although some achievements havebeen made on the uncertainty existing in hydrodynamicmodel of supercavitating vehicle and external perturbationin above researches it is inevitable to ignore many nonlinearfactors in the process of linearization which brings somelimitations to actual application and the particularity ofcontrol object namely supercavitating vehicle cannot bedescribed accurately due to error of modeling In the currentwork the robustness of supercavitating vehicle is studiedwithnonlinear planning force taken into consideration

On the other hand the studies on multistability analysisare mainly made on phase-locked-loop circuits models [1415] electrical machines models [16 17] and aircraft models[18 19] to disclose causes why the systems are susceptible tointrinsic parameters which is helpful to improve efficiency oftransformations in circuits and prevent aircraft from crash-ing with great significance to stability design and utilization

Fns

Fplaning Fgravity

Fcavitator

q

V

w

Figure 2 Structure and forces of supercavitating vehicle

of chaos However the above nonlinear models are all low-dimensional while the model of supercavitating vehicle isdifferent from others which is 4-dimensional in this paperFurthermore aircraftsmove only in the air while underwatersupercavitating vehicles move at high speed inside supercav-ityThe liquid-vapor interface between supercavity and waterflows instability resulting in complicated cavity dynamicsthe nonlinear forces that the vehicle experiences are strongerthan aircraft Due to the complexity of underwater envi-ronment the motion stability is affected by more systemparameters which are coupled and constrained with eachother increasing the complexity of the model so it alwaysleads to unexpected motion the system dynamics presentschallenges to stabilization and the motion robustness of thebody In order to solve this problem it is necessary to makea thorough study on unpredictable factors of the system toprevent unexpected damage Therefore it is significance tostudy the motion stability through multistability analysis

In the paper various coexisting attractors can be foundby adjusting initial values of the system based on a 4-dimensional nonlinear hydrodynamicmodel of supercavitat-ing vehicle after systematic parameters are determined Seenfrom angle of engineering estimation of basins of attractionenables us to know which perturbations are acceptable to thesystem and which (leading to instability of supercavitatingvehicle) are not On such basis the relation between robust-ness of vehicle and the size and change of basins of attractioncan be analyzed further which provides parameters andinitial values for launching stable supercavitating vehicle aswell as necessary base of research for the design of controllerof supercavitating vehicle

2 Dynamic Description of UnderwaterSupercavitating Vehicle

The structure and forces of supercavitating vehicle are shownin Figure 2 supercavitating vehicle is a rotating body the headis cavitator the forepart is a frustum the middle part is acolumn and the tail is a stretching apron-like fin When thevehicle is fully contained in the cavity the only hydrodynamicforces acting are due to the cavitator and immersed finsMainforces exerted on supercavitating vehicle are the lift on thecavitator 119865cavitator the lift on the fin 119865fins the gravity in thecenter of supercavitating vehicle 119865gravity and the nonlinearplanning force 119865planing

The lift on the cavitator is approximately [4]

119865cavitator = 1212058712058811987721198991198812119862119909120572119888 (1)

Advances in Mathematical Physics 3

In the above equation 119862119909 denotes the cavitator dragcoefficient and119862119909 = 1198621199090(1+120590) where1198621199090 = 082 120572119888 denotesthe angle of attack due to cavitator deflection 120575119888 and verticalvelocity 119881 Similarly the lift on the fin is approximately [4]119865fin = minus1198991212058712058811987721198991198812119862119909120572119891 (2)

In (2) the parameter 119899 represents the efficiency of the finwhich is the ratio of the length of the fin immersed in thewater to the total length of the fin and 120572119891 denotes the angle ofattack due to fin deflection 120575119891 and vertical velocity 119881

When the vehicle body navigates in supercavity due tothe change of relative position of the body and the cavitythe tail and the cavity will touch with each other which willproduce a complex nonlinear planning force resulting invibration and shock the normalized force 119865planing is [4]

119865planing = minus1198812 [1 minus ( 1198771015840ℎ119877 + 1198771015840)2]( 1 + ℎ1 + 2ℎ)120572 (3)

In the previous equation 1198771015840 = 119877119888 minus 119877 119877119888 denotes theradius of the cavity and 119877 denotes radius of vehicle Theimmersion depth ℎ and the angle of attack 120572 in the planningforce are given by [4]

ℎ = 0 |119908| lt 119908th = (119877119888 minus 119877)119881119871119871 |119908|119877119881 minus 119877119888 minus 119877119877 otherwise

120572 = 119908 minus 119888119881 119908119881 gt 0119908 + 119888119881 otherwise

(4)

In (4)119908 is the vertical velocity119871 is the vehicle length and119888 is the cavity radius contraction rateDzielski and Kurdila [4] presented a simplified 4-dimen-

sional dynamic model of supercavitating vehicle Althoughonly the effect of angle of attack on planning force isconsidered in the model motion characteristics of supercav-itating vehicle in vertical plane can be described qualitativelyFollowing the work of Dzielski and Kurdila in present worka dynamic model of supercavitating vehicle is established bytwo bifurcation parameters

The center of top surface of the disk-shaped cavitatoron head of supercavitating vehicle is taken as the originof coordinate system The four state variables are used todescribe dynamic of supercavitating vehicle in the modelnamely 119911 119908 120579 and 119902 wherein 119911 represents the depth wherethe body is located 119908 is the vertical velocity and 120579 and 119902refer to the pitch and pitch rate respectively The verticalvelocity119908 is perpendicular to the axial line of supercavitatingvehicle and forward velocity 119881 is parallel to the axial lineIn addition the system has two control inputs namelycavitator deflection angle 120575119888 and fin deflection angle 120575119890 Inthe classic control law presented by Dzielski and Kurdila [4]120575119890 = 0 and 120575119888 = 15119911-30120579-03119902 However supercavitating

vehicle would lack supportive force of fin due to 120575119890 = 0If the lift on the cavitator 119865cavitator could not overcome theweight of supercavitating vehicle vehicle would immerseinto supercavity result in unstable motion Therefore thecontrol law in the paper is chosen as 120575119890 = minus119896119911 1205751198881 =15119911-30120579-03119902 in which 119896 refers to the feedback control gainof 119911 According to fluid dynamics exerted on different partsof supercavitating vehicle the dynamic model [7] can beestablished with cavitation number 120590 and feedback controlgain of fin deflection angle 119896 as variable parameters

(120579119902) = (0 1 minus119881 00 11988622 0 119886240 0 0 10 11988642 0 11988644)(119911119908120579119902)

+( 0 011988721 119887220 011988741 11988742)(120575119890120575119888) +(0119888200)

+( 0119889201198894)119865planing

(5)

In (5)

11988622 = 119862119881119879119898 (minus1 minus 119899119871 ) 119878 + 173611989911987111988624 = 119881119879119878 (119862minus119899119898 + 79) minus 119881119879(119862minus119899119898 + 1736) 1736119871211988642 = 119862119881119879119898 (1736 minus 1111989936 ) 11988644 = minus1111986211988111987911989911987136119898 11988721 = 1198621198812119879119899119898 (minus119878119871 + 1711987136 ) 11988722 = minus1198621198812119879119878119898119871 11988741 = minus11119862119881211987911989936119898 11988742 = 17119862119881211987936119898 1198882 = 1198921198892 = 119879119898 (minus1711987136 + 119878119871)

4 Advances in Mathematical Physics

Table 1 Parameters of supercavitating vehicle model

Parameter Description Value119892 Acceleration of gravity 981ms2

m Density ratio (120588119898120588) 2n Tail efficiency 05119877119899 Radius of cavitator 00191mR Radius of vehicle 00508mL Length of vehicle 18m120590 Cavitation number [001980 003680]1198621199090 Coefficient of lift 082

0024 0028 0032 00360020

minus80

minus60

minus40

minus20

0

20

40

120590

k

Figure 3 Distribution diagram of dynamic behaviors of supercavi-tating vehicle

1198894 = 1111987936119898119878 = 11601198772 + 1331198712405 119879 = 171198789 minus 28911987121296 119862 = 051198621199090 (1 + 120590) (119877119899119877 )2

(6)119898 (120588119898120588) denotes density ratio where 120588119898 is the specificationof a uniform density for the vehicle and 120588 is the density ofwater

3 General Dynamic Characteristics ofSupercavitating Vehicle

Values of parameters for supercavitating vehicle model aregiven in Table 1 [7] Based on dynamic model (5) the initialconditions are selected randomly According to the Lyapunovstability theory the stable solutions periodic solutions andchaotic solutions of the model are represented by greenred and black in Figure 3 respectively The distributiondiagram of dynamic behaviors defined by cavitation number120590 and feedback control gain of fin deflection angle 119896 as

the bifurcation parameters is drawn And the dependenceof dynamic behaviors on 120590 and 119896 is described As shownin Figure 3 if value of 120590 119896 is within the green regionthe max Lyapunov exponent of (5) is a negative and statevariables 119911 119908 120579 and 119902 converge on stable equilibriumpoints supercavitating vehicle can move steadily If the point(120590 119896) is within the red region the max Lyapunov exponentof (5) is zero and state variables 119911 119908 120579 and 119902 oscillateperiodically centering on stable equilibriumpoints thereforesupercavitating vehicle shock periodically If (120590 119896) is withinthe black region the max Lyapunov exponent of (5) is apositive and state variables 119911 119908 120579 and 119902 oscillate violentlyand irregularly violent vibrations and shocks will occur andthen supercavitating vehicle will capsizeThe light blue regionrepresents where the system is divergent and vehicle cannotnavigate

Figure 3 reflects different dynamics of supercavitatingvehicle completely when the parameters 120590 and 119896 changesimultaneously The range of parameters corresponding tostable motion of vehicle can be determined by the dynamicdistribution diagramWhen cavitation number 120590 is constantthe value of feedback control gain of fin deflection angle 119896can be adjusted within the green stable region and the redperiodic region to realize stable motion of supercavitatingvehicle efficiently which is instructive to stability control ofsupercavitating vehicle It can be seen from Figure 3 that onehas the following

(1) The horizontal section is bifurcation diagram of thesystem varying with 120590 while the vertical section isbifurcation diagram of the system varying with 119896Thebifurcation diagram reflects the rules of changes ofthe system with parameters and complex nonlinearphysical phenomena generatedThe Hopf bifurcationalways occurs when the system switches from steadystate to periodic state so the boundary betweenthe red region and the green region in Figure 3 iscritical line between steady state and periodic statealso called the Hopf bifurcation line The bound-ary between the red region and the black regionrepresents the switching between the periodic stateand the chaotic state There are nonlinear physicalphenomena such as the cutting bifurcation or period-doubling bifurcation at this boundary

(2) Supercavitating vehicle has three stable states includ-ing stable motion periodic motion and chaotic stateSelect one point from the three regions respectivelythe projections of phase tracks on 119908-120579 plane areshown in Figure 4 Select 120590 = 002249 119896 = minus1405from the green region as shown in Figure 4(a)vertical velocity119908 and pitch 120579 are attracted to a stableequilibrium point functioned by feedback controllaw supercavitating vehicle is exerted by forces inequilibrium state navigating stably in fixed positionand attitude inside supercavity with oblique smallangle of attack Select 120590 = 003459 119896 = minus175 ran-domly from the red region as shown in Figure 4(b)mapping of the system forms a closed limit cycle andthe limit cycle intersects the red switching critical line

Advances in Mathematical Physics 5

minus001

0

001

015 020010

120579(rad)

w (mmiddotsminus1)

(a)

minus001

0

001

002

0 5minus5

120579(rad)

w (mmiddotsminus1)

(b)

minus0004

minus0002

0

0002

0004

minus1 0 1 2minus2

120579(rad)

w (mmiddotsminus1)

(c)

Figure 4 Phase tracks of three stable states on 119908-120579 plane (a) 120590 = 002249 119896 = minus1405 (b) 120590 = 003459 119896 = minus175 (c) 120590 = 003040119896 = minus5913119908 = 119908th where 119908th = (119877119888 minus 119877)119881119871 [7] 119877119888 is theradius of supercavity and vertical velocity 119908 fluc-tuates around 119908th Tail of supercavitating vehicleoscillates from time to time touching the supercavitywhich leads to periodic change of planning forcesometimes the tail penetrates the supercavity andinserts into water to generate planning force andsometimes it is enclosed by the supercavity and thusno planning force is generated and supercavitatingvehicle oscillates periodically Select parameters 120590 =003040 119896 = minus5913 randomly from the red regionappearance of chaotic attractor indicates that super-cavitating vehicle has complex nonlinear dynamicbehavior and is likely to capsize In practical applica-tion of engineering effective control should be takento prevent occurrence of such circumstance

(3) Within 120590 isin [00262 002973] 119896 isin [7387 1432] thegreen stable region the red periodic region and thedivergent region are interwoven stable equilibriumpoints coexist with periodic attractors in the regionWhen 120590 = 002771 119896 = 7883 if initial values are(1199110 1199080 1205790 1199020) = (049 103 073 minus031) the phase

track converges to a stable equilibrium point If initialvalues are (1199110 1199080 1205790 1199020) = (029 minus079 089 minus115)the phase track is a limit cycle Projections of thecoexisting attractor in two-dimensional plane 119908-120579and three-dimensional space119908-120579-119902 are shown in Fig-ures 5(a) and 5(b) respectively in which the red dotrepresents equilibrium point attractor and the bluelimit cycle represents periodic attractor Within 120590 isin[002745 003255] 119896 isin [minus7685 minus5257] periodicstate always scatters in chaotic region and periodicattractors coexist with chaotic attractors in the dottedregion When 120590 = 003259 119896 = minus5604 if initial val-ues are (1199110 1199080 1205790 1199020) = (minus019 089 minus076 minus141)the phase track converges to a periodic attractorwhile if initial values are (1199110 1199080 1205790 1199020) = (minus106235 minus062 075) the phase track converges to achaotic attractor coexisting attractors are shown inFigures 5(c) and 5(d) the periodic attractor is markedwith red limit cycle and the chaotic attractor ismarked with blue In addition there are other typesof multistability in the system at several combinationsof parameters such as coexistence of different stable

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Figure 5 Projections of coexisting attractors on 119908-120579 plane and 119908-120579-q spaceTable 2 Coexisting attractors at different combinations of parameters

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

120590 = 002468119896 = 9011 (084 minus089 010 minus054)(minus142 049 minus018 minus019) (minus555 minus623 minus8648 minus8697)(minus829 minus901 minus6341 minus6392) 00

Coexistence of stable equilibrium pointwith periodic attractor

120590 = 002603119896 = 5901 (032 minus131 minus043 034)(054 183 minus226 086) (minus1188 minus1212 minus2846 minus2909)(0014 minus1247 minus1783 minus5046) 01

Coexistence of different 1-periodic cycles 120590 = 002805119896 = 6961 (054 183 minus226 086)(minus012 149 141 142) (minus004 minus1165 minus1686 minus6567)(001 minus1122 minus1595 minus5375) 11

Coexistence of 1-periodic cycle with2-periodic cycle

120590 = 003153119896 = minus7313 (144 minus196 minus020 minus121)(030 minus060 049 074) (002 minus214 minus2187 minus7592)(minus002 minus208 minus2115 minus7601) 11

Coexistence of periodic attractor withchaotic attractor

120590 = 003238119896 = minus6263 (minus109 003 055 110)(minus077 037 minus023 112) (003 minus2116 minus2905 minus2990)(231 minus064 minus8520 minus9403) 1232

equilibrium points and coexistence of multiple limitcycles When 120590 = 003171 119896 = minus71432 as shown inFigures 5(e) and 5(f) a red 1-periodic cycle coexistswith a blue 2-periodic cycle When 120590 = 002805 119896 =6961 as shown in Figures 5(g) and 5(h) two differentkinds of 1-periodic cycles coexist When 120590 = 002468119896 = 9011 as shown in Figures 5(i) and 5(j) twodifferent kinds of stable equilibrium points coexistIf parameters are invariable the coexisting attractorsindicate behaviors of the system sensitive to initialconditions the trajectory of supercavitating vehiclemay probably approach two types of attractors whenits initial depth vertical velocity pitch and pitch rateare taken different values namely motion state ofsupercavitating vehicle is likely to be different

(4) Within 120590 isin [00198 002956] 119896 isin [7883 1779] thegreen stable dots scatter in the blue divergent regionWhen 119896 isin [minus95 minus8578] red periodic dots scatterwithin the blue divergent region In the interwovenregions slight change in parameters can always leadto change in motion state of supercavitating vehicleimproper setting of initial conditions would makesupercavitating vehicle capsize Basins of attraction atstable equilibrium points and periodic dots are not

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

It can be derived from above analysis that various coexist-ing attractors exist in the parameter regions marked withdifferent colors in Figure 3 as shown in Table 2 As longas the coexisting attractors and their types are known eachattractor can be associated with all the initial conditions thatmake its trajectory approach this attractor they constituteclustering region of the attractor which is called the basinsof attraction [14 15] So that the final state of the systemis determined by the basins of attraction where initialconditions are located when the initial conditions are nearthe boundary of the attraction basins slight disturbanceor change in parameters will lead to a completely differentmotion of the system The initial conditions will bring highuncertainty in the final destination of the system as welland complex behaviors [16 17] often appear which arelikely to deviate from original expectation of designer andlead to unpredictable motion and thus pose a great threatto the engineering application Therefore it is essential to

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 6 Basins of attraction (a) 120590 = 002468 119896 = 3423 (b) 120590 = 002603 119896 = 5901 (c) 120590 = 002721 119896 = 7473 (d) 120590 = 002655 119896 = 9369understand such abnormal phenomena [18] thoroughly Forcomprehensive analysis of a complex system complexity ofits attractors and its basins of attraction should be analyzedsimultaneously The study of basins of attraction is of highvalue in engineering application Most engineering problemsinvolve not only analysis of local stability and bifurcationunder minor perturbation but also scope of the attractionbasins of steady solutions that is the area of attraction basinswhere steady solution is of same properties With differentlaunching conditions the larger the area of attraction basinsat steady state and periodic state the stronger the robustnessof supercavitating vehicle

41 Basins of Attraction of Stable Equilibrium Point and LimitCycle Select various combinations of parameters from 120590 isin[002620 002973] 119896 isin [7387 14325] namely the regioninterwoven by stable periodic and divergent states thesections of basins of attraction on 1199110-1205790 plane are shownin Figure 6 and basins of attraction are approximate to aparallelogram when launching depth 1199110 and launching pitch1205790 are selected from green region in the figure the lift ofcavitator and fin is equal to the weight of supercavitatingvehicle which make supercavitating vehicle move stablyunder the parameters When initial values are correspondingto the dots within red region tail of supercavitating vehicle

oscillates periodically moving into and out of the super-cavity alternatively When initial values are correspondingto the dots within blue region too high values of initialconditions would lead to divergence of the system and theincapability of supercavitating vehicle to navigate If anyphase dot within basins of attraction of certain attractor istaken as initial condition the system always converges tosuch attractor In combinations of parameters if cavitationnumber varies within a small range the feedback controlgain of fin deflection angle 119896 decreases gradually it can beproven by calculating the area of corresponding attractionbasins that 1198781198861 gt 1198781198871 gt 1198781198881 gt 1198781198891 1198781198861 1198781198871 1198781198881 and 1198781198891 areareas of attraction basins in Figures 6(a) 6(b) 6(c) and6(d) respectivelyTherefore provided that cavitation numbervaries within a small range the area of attraction basinsdecreases with the feedback control gain of fin deflectionangle 119896 With target to different initial launching conditionsthe lower the value of 119896 the smaller the attraction basinof stable equilibrium point and periodic attractor and theweaker the robustness of the system and hence supercav-itating vehicle is more sensitive to external interferencesand becomes unstable In addition when values of initialconditions are corresponding to the dots close to boundariesof the red region and the green region slight interferencecan lead to the switching between stable motion and periodic

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

Figure 7 Basins of attraction (a) 120590 = 002838 119896 = minus5901 (b) 120590 = 003136 119896 = minus6744 (c) 120590 = 003238 119896 = minus6263 (d) 120590 = 003259119896 = minus5604oscillation of the vehicle In practical engineering applicationrobustness of supercavitating vehicle can be improved byadjusting fin deflection angel of supercavitating vehicle andinitial launching conditions

42 Basins of Attraction of Limit Cycle and Chaotic AttractorSelect a few dots from 120590 isin [002745 003255] 119896 isin [minus76851minus52573] namely the regionwhere periodic state interweaveswith chaotic state sections of their basins of attraction on1199110-1205790 plane are shown in Figure 7 the red region representsthe initial values falling into periodic trajectory and super-cavitating vehicle oscillates periodically The black regionrepresents the initial values which draw supercavitating vehi-cle to chaotic state eventually supercavitating vehicle thusbecomes unstable or even capsizesTheblue region representsthe initial values which make the system divergent InFigure 7 under different combinations of parameters basinsof attraction are almost same in shape butwith different sizesThere is a fractal boundary in which slight change in initialconditions may probably trigger the transition of the systemfrom stable periodic state to chaotic state and often leadto weaker robustness and even capsizing of supercavitatingvehicle In practical engineering application it should be

avoided to choose such variable combinations of parametersso as to improve robustness of the system

43 Basins of Attraction of Stable Equilibrium Points Selectseveral dots from 120590 isin [001980 002956] 119896 isin [7883 17794]namely the region where stable state interweaves with diver-gent region Sections of their basins of attraction on 1199110-1205790plane are shown in Figure 8 If initial values 1199110 and 1205790 arewithin green region supercavity can be formed to enclosevehicle and supercavitating vehicle moves stably When theinitial values are within the blue region rather than the greenregion supercavitating vehicle becomes unstable The biggerthe basins of attraction the stronger the robustness of thesystem It can be seen from Figure 8 that if cavitation numbervaries within small range the feedback control gain of findeflection angle 119896 decreases gradually and the area of attrac-tion basins of stable equilibrium points varies with combina-tions of parameters 1198781198862 gt 1198781198872 gt 1198781198882 gt 1198781198892 where 1198781198862 1198781198872 1198781198882and 1198781198892 are area of attraction basins in Figures 8(a) 8(b)8(c) and 8(d) respectively The robustness decreases in suchorder Therefore within such scope provided that cavitationnumber varies within a small range the lower the value of119896 the bigger the area of attraction basins and the stronger

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 8 Basins of attraction (a) 120590 = 002788 119896 = 8874 (b) 120590 = 002838 119896 = 9369 (c) 120590 = 002754 119896 = 1106 (d) 120590 = 002636 119896 = 1004the robustness supercavitating vehicle is more liable to movestably Figure 8 not only shows the range of initial conditionsfor stable motion of supercavitating vehicle but also reflectsthe relationship between the robustness and the parametersIn practical application parameters with bigger basins ofattraction and strong robustness should be selected to ensurestable motion of supercavitating vehicle

44 Basins of Attraction of Limit Cycle Select several dotsfrom 119896 isin [minus89895 minus85786] namely the region where peri-odic state interweaves with stable state Sections of theirbasins of attraction on 1199110-1205790 plane are shown in Figure 9If initial values 1199110 and 1205790 are within the red region peri-odic oscillation occurs due to collision between the tail ofsupercavitating vehicle and the supercavity from time totime supercavitating vehicle is in periodic oscillation Whenthe initial values are corresponding to the dots within theblue region the system is divergent and supercavitatingvehicle capsizes It can be seen from Figure 9 that providedthat cavitation number varies within small range the areaof attraction basins decreases with value of 119896 graduallyTherefore value of 119896 has significant effect on robustness ofsupercavitating vehicle which becomes weaker with increaseof 119896

45 Basins of Attraction of Other Types It can be found fromanalysis of dynamic distribution diagram that coexistence ofstable equilibrium point with limit cycle and coexistence oflimit cycle with chaotic attractor are universal phenomena ofmultistability In addition there are other types ofmultistabil-ity at individual parameters in the system such as coexistenceof 1-periodic cycle with 2-periodic cycle coexistence of twotypes of 1-periodic cycles and coexistence of two types ofstable equilibrium points For the coexistence of 1-periodiccycle with 2-periodic cycle sections of basins of attraction on1199110-1205790 plane are shown in Figures 10(a) and 10(b) respectivelyThe red region represents the initial values which makesupercavitating vehicle approach the 1-periodic trajectoryeventually The dark blue region represents the initial valueswhich make vehicle approach 2-periodic trajectory eventu-allyThe light blue region represents divergence of the systemTherefore different initial launching conditions may proba-bly result in different periods of motion Basin of attractionfor coexistence of two types of 1-periodic cycles is shown inFigure 10(c) the red region and the dark blue region representthe initial values which make supercavitating vehicle fallinto different periods of oscillation eventually With suchcombinations of parameters supercavitating vehicle maydisplay different periodic microoscillation under different

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

Figure 9 Basins of attraction (a) 120590 = 002434 119896 = minus8783 (b) 120590 = 003040 119896 = minus9193 (c) 120590 = 002805 119896 = minus909 (d) 120590 = 003428119896 = minus8988initial conditions Basin of attraction for coexistence of twotypes of stable equilibrium points is shown in Figure 10(d)when initial values correspond with the red region and thedark blue region the motion of supercavitating vehicle willapproach two different types of stable equilibrium pointsSupercavitating vehicle can navigate stably if initial values arewithin the basin of attractionThe system is divergent and thevehicle capsizes once initial values are beyond the boundaryof the basin of attraction

5 Conclusion

In this paper general characteristics of dynamic behaviorsof supercavitating vehicle are studied with dynamic distri-bution diagram based on a 4-dimensional dynamic modelcoexistence of various attractors is confirmed herein and therelation between the robustness of supercavitating vehicleand the parameters of system and initial values is obtainedthough multistability analysis Following conclusions can bedrawn

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

dynamic distribution diagram the relationshipbetween dynamic behaviors and variable parameterscan be obtained And the regions of parameterswhere coexisting attractors are likely to be found aredisclosed providing the basis for setting parametersfor stable motion

(2) Basins of attraction vary with parameters of thesystem some attractors will die and new attractorswill be generated Basins of attraction can be usedto determine the range of initial conditions for sta-ble motion of supercavitating vehicle and unstablemotion can be prevented by adjusting initial values oflaunch

(3) Generally supercavitating vehicle has three stablestates including stable periodic and chaotic statesUnder appropriate combinations of parameters thereare various motions of multistability such as coex-istence of two types of stable equilibrium pointscoexistence of a stable equilibrium point with alimit cycle coexistence of a limit cycle with chaoticattractor and coexistence of multiple limit cycles

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Figure 10 Basins of attraction (a) 120590 = 003171 119896 = minus7143 (b) 120590 = 003187 119896 = minus7094 (c) 120590 = 002805 119896 = 6961 (d) 120590 = 002464119896 = 9011When parameters are fixed supercavitating vehiclemay display different states of motion under differentinitial values

(4) Provided that cavitation number varies within a smallrange robustness of the system becomes weaker withthe increase of feedback control gain of fin deflectionangle 119896 size of basins of attraction becomes smallerand robustness of the system becomes weaker Sys-tematic parameters with greater basins of attractioncan be selected to lessen sensitivity to external inter-ference and improve robustness of supercavitatingvehicle

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant nos 11402116 and 11472163) and

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

[1] A Ducoin B Huang and Y L Young ldquoNumerical modeling ofunsteady cavitating flows around a stationary hydrofoilrdquo Inter-national Journal of Rotating Machinery vol 2012 Article ID215678 17 pages 2012

[2] B Vanek J Bokor G J Balas and R E A Arndt ldquoLongitudinalmotion control of a high-speed supercavitation vehiclerdquo Journalof Vibration and Control vol 13 no 2 pp 159ndash184 2007

[3] Q-T Li Y-S He L-P Xue and Y-Q Yang ldquoA numericalsimulation of pitching motion of the ventilated supercavitingvehicle around its noserdquo Chinese Journal of Hydrodynamics Avol 26 no 6 pp 689ndash696 2011

[4] J Dzielski and A Kurdila ldquoA benchmark control problem forsupercavitating vehicles and an initial investigation of solu-tionsrdquo Journal of Vibration andControl vol 9 no 7 pp 791ndash8042003

[5] M A Hassouneh V Nguyen B Balachandran and E H AbedldquoStability analysis and control of supercavitating vehicles with

Advances in Mathematical Physics 13

advection delayrdquo Journal of Computational and Nonlinear Dy-namics vol 8 no 2 Article ID 021003 10 pages 2013

[6] B Vanek J Bokor and G Balas ldquoTheoretical aspects of high-speed supercavitation vehicle controlrdquo American Control Con-ference vol 6 no 3 pp 1ndash4 2006

[7] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007

[8] G Lin B Balachandran and E H Abed ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASME Inter-national Mechanical Engineering Congress and Exposition pp293ndash300 Chicago Ill USA November 2006

[9] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008

[10] B Vanek andG Balas ldquoControl of high-speed underwater vehi-clesrdquo Control of Uncertain Systems Modeling Approximationand Design no 329 pp 25ndash44 2006

[11] A Goel Robust control of supercavitating vehicles in the presenceof dynamic and uncertain cavity [Doctorrsquos thesis] University ofFlorida Gainesville Fla USA 2005

[12] X H Zhao Y Sun Z K Qi and M Y Han ldquoCatastrophecharacteristics and control of pitching supercavitating vehiclesat fixed depthsrdquo Ocean Engineering vol 112 pp 185ndash194 2016

[13] M L Wang and G L Zhao ldquoRobust controller design forsupercavitating vehicles based on BTT maneuvering strategyrdquoin Proceedings of the International Conference on Mechatronicsamp Automation pp 227ndash231 2007

[14] G A Leonov and N V Kuznetsov ldquoHidden attractors in dy-namical systems from hidden oscillations in hilbert-kol-mogorov Aizerman and Kalman problems to hidden chaoticattractor in chua circuitsrdquo International Journal of Bifurcationand Chaos vol 23 no 1 Article ID 1330002 2013

[15] M Chen M Li Q Yu B Bao Q Xu and J Wang ldquoDynamicsof self-excited attractors and hidden attractors in generalizedmemristor-based Chuarsquos circuitrdquo Nonlinear Dynamics vol 81no 1-2 pp 215ndash226 2015

[16] J C A de Bruin A Doris N van de Wouw W P M HHeemels and H Nijmeijer ldquoControl of mechanical motionsystems with non-collocation of actuation and friction a Popovcriterion approach for input-to-state stability and set-valuednonlinearitiesrdquo Automatica vol 45 no 2 pp 405ndash415 2009

[17] M A Kiseleva N V Kuznetsov G A Leonov and P Neit-taanmaki ldquoDrilling systems failures and hidden oscillationsrdquoin Proceedings of the IEEE 4th International Conference onNonlinear Science and Complexity (NSC rsquo12) pp 109ndash112 IEEEBudapest Hungary August 2012

[18] G A Leonov N V Kuznetsov O A Kuznetsova SM Seledzhiand V I Vagaitsev ldquoHidden oscillations in dynamical systemsrdquoWSEAS Transactions on Systems and Control vol 6 no 2 pp54ndash67 2011

[19] T Lauvdal R M Murray and T I Fossen ldquoStabilizationof integrator chains in the presence of magnitude and ratesaturations a gain scheduling approachrdquo in Proceedings of the36th IEEE Conference on Decision and Control pp 4004ndash4005December 1997

Submit your manuscripts athttpswwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

Figure 1 Illustration of vehicle inside supercavity

to ensure the stable motion of underwater supercavitatingvehicle

Unique navigation environment of supercavitating vehi-cle has predetermined that it is a multivariable system withcomplex couplings and uncertain hydrodynamic parametersSupercavitating vehicle is different fromcommonunderwatervehicle both particularities of control object such as nonlin-earity of the planning force anduncertainty of supercavitatingvehicle as well as the robustness with external interferencesshould be taken into consideration There are a few paperspublished regarding the study of robustness Linear state-feedback controlmethod presented by Lin et al [9] achievedstrong robustness in error of modeling for planning forceof supercavitating vehicle Vanek and Balas [10] considereduncertainty in shape of supercavity and the control withlinear parameter variables (ldquoLPVrdquo) after linearization offeedback was researched Goel [11] applied robustness controlto the design and analysis of linear model of supercavitatingvehicle with target to uncertainty of its hydrodynamic param-eters Zhao et al [12] considered the uncertainty (existingin linear model of supercavitating vehicle) and unmodeleddynamics (mainly displayed by hydrodynamic coefficientperturbation in simulation) and applied robustness control tothe design of controllerWang andZhao [13] applied principleof minor perturbation to linearization of longitudinal motionmodel of supercavitating vehicle and control methods werefurther researched based on linear feedback theory androbustness control theory Although some achievements havebeen made on the uncertainty existing in hydrodynamicmodel of supercavitating vehicle and external perturbationin above researches it is inevitable to ignore many nonlinearfactors in the process of linearization which brings somelimitations to actual application and the particularity ofcontrol object namely supercavitating vehicle cannot bedescribed accurately due to error of modeling In the currentwork the robustness of supercavitating vehicle is studiedwithnonlinear planning force taken into consideration

On the other hand the studies on multistability analysisare mainly made on phase-locked-loop circuits models [1415] electrical machines models [16 17] and aircraft models[18 19] to disclose causes why the systems are susceptible tointrinsic parameters which is helpful to improve efficiency oftransformations in circuits and prevent aircraft from crash-ing with great significance to stability design and utilization

Fns

Fplaning Fgravity

Fcavitator

q

V

w

Figure 2 Structure and forces of supercavitating vehicle

of chaos However the above nonlinear models are all low-dimensional while the model of supercavitating vehicle isdifferent from others which is 4-dimensional in this paperFurthermore aircraftsmove only in the air while underwatersupercavitating vehicles move at high speed inside supercav-ityThe liquid-vapor interface between supercavity and waterflows instability resulting in complicated cavity dynamicsthe nonlinear forces that the vehicle experiences are strongerthan aircraft Due to the complexity of underwater envi-ronment the motion stability is affected by more systemparameters which are coupled and constrained with eachother increasing the complexity of the model so it alwaysleads to unexpected motion the system dynamics presentschallenges to stabilization and the motion robustness of thebody In order to solve this problem it is necessary to makea thorough study on unpredictable factors of the system toprevent unexpected damage Therefore it is significance tostudy the motion stability through multistability analysis

In the paper various coexisting attractors can be foundby adjusting initial values of the system based on a 4-dimensional nonlinear hydrodynamicmodel of supercavitat-ing vehicle after systematic parameters are determined Seenfrom angle of engineering estimation of basins of attractionenables us to know which perturbations are acceptable to thesystem and which (leading to instability of supercavitatingvehicle) are not On such basis the relation between robust-ness of vehicle and the size and change of basins of attractioncan be analyzed further which provides parameters andinitial values for launching stable supercavitating vehicle aswell as necessary base of research for the design of controllerof supercavitating vehicle

2 Dynamic Description of UnderwaterSupercavitating Vehicle

The structure and forces of supercavitating vehicle are shownin Figure 2 supercavitating vehicle is a rotating body the headis cavitator the forepart is a frustum the middle part is acolumn and the tail is a stretching apron-like fin When thevehicle is fully contained in the cavity the only hydrodynamicforces acting are due to the cavitator and immersed finsMainforces exerted on supercavitating vehicle are the lift on thecavitator 119865cavitator the lift on the fin 119865fins the gravity in thecenter of supercavitating vehicle 119865gravity and the nonlinearplanning force 119865planing

The lift on the cavitator is approximately [4]

119865cavitator = 1212058712058811987721198991198812119862119909120572119888 (1)

Advances in Mathematical Physics 3

In the above equation 119862119909 denotes the cavitator dragcoefficient and119862119909 = 1198621199090(1+120590) where1198621199090 = 082 120572119888 denotesthe angle of attack due to cavitator deflection 120575119888 and verticalvelocity 119881 Similarly the lift on the fin is approximately [4]119865fin = minus1198991212058712058811987721198991198812119862119909120572119891 (2)

In (2) the parameter 119899 represents the efficiency of the finwhich is the ratio of the length of the fin immersed in thewater to the total length of the fin and 120572119891 denotes the angle ofattack due to fin deflection 120575119891 and vertical velocity 119881

When the vehicle body navigates in supercavity due tothe change of relative position of the body and the cavitythe tail and the cavity will touch with each other which willproduce a complex nonlinear planning force resulting invibration and shock the normalized force 119865planing is [4]

119865planing = minus1198812 [1 minus ( 1198771015840ℎ119877 + 1198771015840)2]( 1 + ℎ1 + 2ℎ)120572 (3)

In the previous equation 1198771015840 = 119877119888 minus 119877 119877119888 denotes theradius of the cavity and 119877 denotes radius of vehicle Theimmersion depth ℎ and the angle of attack 120572 in the planningforce are given by [4]

ℎ = 0 |119908| lt 119908th = (119877119888 minus 119877)119881119871119871 |119908|119877119881 minus 119877119888 minus 119877119877 otherwise

120572 = 119908 minus 119888119881 119908119881 gt 0119908 + 119888119881 otherwise

(4)

In (4)119908 is the vertical velocity119871 is the vehicle length and119888 is the cavity radius contraction rateDzielski and Kurdila [4] presented a simplified 4-dimen-

sional dynamic model of supercavitating vehicle Althoughonly the effect of angle of attack on planning force isconsidered in the model motion characteristics of supercav-itating vehicle in vertical plane can be described qualitativelyFollowing the work of Dzielski and Kurdila in present worka dynamic model of supercavitating vehicle is established bytwo bifurcation parameters

The center of top surface of the disk-shaped cavitatoron head of supercavitating vehicle is taken as the originof coordinate system The four state variables are used todescribe dynamic of supercavitating vehicle in the modelnamely 119911 119908 120579 and 119902 wherein 119911 represents the depth wherethe body is located 119908 is the vertical velocity and 120579 and 119902refer to the pitch and pitch rate respectively The verticalvelocity119908 is perpendicular to the axial line of supercavitatingvehicle and forward velocity 119881 is parallel to the axial lineIn addition the system has two control inputs namelycavitator deflection angle 120575119888 and fin deflection angle 120575119890 Inthe classic control law presented by Dzielski and Kurdila [4]120575119890 = 0 and 120575119888 = 15119911-30120579-03119902 However supercavitating

vehicle would lack supportive force of fin due to 120575119890 = 0If the lift on the cavitator 119865cavitator could not overcome theweight of supercavitating vehicle vehicle would immerseinto supercavity result in unstable motion Therefore thecontrol law in the paper is chosen as 120575119890 = minus119896119911 1205751198881 =15119911-30120579-03119902 in which 119896 refers to the feedback control gainof 119911 According to fluid dynamics exerted on different partsof supercavitating vehicle the dynamic model [7] can beestablished with cavitation number 120590 and feedback controlgain of fin deflection angle 119896 as variable parameters

(120579119902) = (0 1 minus119881 00 11988622 0 119886240 0 0 10 11988642 0 11988644)(119911119908120579119902)

+( 0 011988721 119887220 011988741 11988742)(120575119890120575119888) +(0119888200)

+( 0119889201198894)119865planing

(5)

In (5)

11988622 = 119862119881119879119898 (minus1 minus 119899119871 ) 119878 + 173611989911987111988624 = 119881119879119878 (119862minus119899119898 + 79) minus 119881119879(119862minus119899119898 + 1736) 1736119871211988642 = 119862119881119879119898 (1736 minus 1111989936 ) 11988644 = minus1111986211988111987911989911987136119898 11988721 = 1198621198812119879119899119898 (minus119878119871 + 1711987136 ) 11988722 = minus1198621198812119879119878119898119871 11988741 = minus11119862119881211987911989936119898 11988742 = 17119862119881211987936119898 1198882 = 1198921198892 = 119879119898 (minus1711987136 + 119878119871)

4 Advances in Mathematical Physics

Table 1 Parameters of supercavitating vehicle model

Parameter Description Value119892 Acceleration of gravity 981ms2

m Density ratio (120588119898120588) 2n Tail efficiency 05119877119899 Radius of cavitator 00191mR Radius of vehicle 00508mL Length of vehicle 18m120590 Cavitation number [001980 003680]1198621199090 Coefficient of lift 082

0024 0028 0032 00360020

minus80

minus60

minus40

minus20

0

20

40

120590

k

Figure 3 Distribution diagram of dynamic behaviors of supercavi-tating vehicle

1198894 = 1111987936119898119878 = 11601198772 + 1331198712405 119879 = 171198789 minus 28911987121296 119862 = 051198621199090 (1 + 120590) (119877119899119877 )2

(6)119898 (120588119898120588) denotes density ratio where 120588119898 is the specificationof a uniform density for the vehicle and 120588 is the density ofwater

3 General Dynamic Characteristics ofSupercavitating Vehicle

Values of parameters for supercavitating vehicle model aregiven in Table 1 [7] Based on dynamic model (5) the initialconditions are selected randomly According to the Lyapunovstability theory the stable solutions periodic solutions andchaotic solutions of the model are represented by greenred and black in Figure 3 respectively The distributiondiagram of dynamic behaviors defined by cavitation number120590 and feedback control gain of fin deflection angle 119896 as

the bifurcation parameters is drawn And the dependenceof dynamic behaviors on 120590 and 119896 is described As shownin Figure 3 if value of 120590 119896 is within the green regionthe max Lyapunov exponent of (5) is a negative and statevariables 119911 119908 120579 and 119902 converge on stable equilibriumpoints supercavitating vehicle can move steadily If the point(120590 119896) is within the red region the max Lyapunov exponentof (5) is zero and state variables 119911 119908 120579 and 119902 oscillateperiodically centering on stable equilibriumpoints thereforesupercavitating vehicle shock periodically If (120590 119896) is withinthe black region the max Lyapunov exponent of (5) is apositive and state variables 119911 119908 120579 and 119902 oscillate violentlyand irregularly violent vibrations and shocks will occur andthen supercavitating vehicle will capsizeThe light blue regionrepresents where the system is divergent and vehicle cannotnavigate

Figure 3 reflects different dynamics of supercavitatingvehicle completely when the parameters 120590 and 119896 changesimultaneously The range of parameters corresponding tostable motion of vehicle can be determined by the dynamicdistribution diagramWhen cavitation number 120590 is constantthe value of feedback control gain of fin deflection angle 119896can be adjusted within the green stable region and the redperiodic region to realize stable motion of supercavitatingvehicle efficiently which is instructive to stability control ofsupercavitating vehicle It can be seen from Figure 3 that onehas the following

(1) The horizontal section is bifurcation diagram of thesystem varying with 120590 while the vertical section isbifurcation diagram of the system varying with 119896Thebifurcation diagram reflects the rules of changes ofthe system with parameters and complex nonlinearphysical phenomena generatedThe Hopf bifurcationalways occurs when the system switches from steadystate to periodic state so the boundary betweenthe red region and the green region in Figure 3 iscritical line between steady state and periodic statealso called the Hopf bifurcation line The bound-ary between the red region and the black regionrepresents the switching between the periodic stateand the chaotic state There are nonlinear physicalphenomena such as the cutting bifurcation or period-doubling bifurcation at this boundary

(2) Supercavitating vehicle has three stable states includ-ing stable motion periodic motion and chaotic stateSelect one point from the three regions respectivelythe projections of phase tracks on 119908-120579 plane areshown in Figure 4 Select 120590 = 002249 119896 = minus1405from the green region as shown in Figure 4(a)vertical velocity119908 and pitch 120579 are attracted to a stableequilibrium point functioned by feedback controllaw supercavitating vehicle is exerted by forces inequilibrium state navigating stably in fixed positionand attitude inside supercavity with oblique smallangle of attack Select 120590 = 003459 119896 = minus175 ran-domly from the red region as shown in Figure 4(b)mapping of the system forms a closed limit cycle andthe limit cycle intersects the red switching critical line

Advances in Mathematical Physics 5

minus001

0

001

015 020010

120579(rad)

w (mmiddotsminus1)

(a)

minus001

0

001

002

0 5minus5

120579(rad)

w (mmiddotsminus1)

(b)

minus0004

minus0002

0

0002

0004

minus1 0 1 2minus2

120579(rad)

w (mmiddotsminus1)

(c)

Figure 4 Phase tracks of three stable states on 119908-120579 plane (a) 120590 = 002249 119896 = minus1405 (b) 120590 = 003459 119896 = minus175 (c) 120590 = 003040119896 = minus5913119908 = 119908th where 119908th = (119877119888 minus 119877)119881119871 [7] 119877119888 is theradius of supercavity and vertical velocity 119908 fluc-tuates around 119908th Tail of supercavitating vehicleoscillates from time to time touching the supercavitywhich leads to periodic change of planning forcesometimes the tail penetrates the supercavity andinserts into water to generate planning force andsometimes it is enclosed by the supercavity and thusno planning force is generated and supercavitatingvehicle oscillates periodically Select parameters 120590 =003040 119896 = minus5913 randomly from the red regionappearance of chaotic attractor indicates that super-cavitating vehicle has complex nonlinear dynamicbehavior and is likely to capsize In practical applica-tion of engineering effective control should be takento prevent occurrence of such circumstance

(3) Within 120590 isin [00262 002973] 119896 isin [7387 1432] thegreen stable region the red periodic region and thedivergent region are interwoven stable equilibriumpoints coexist with periodic attractors in the regionWhen 120590 = 002771 119896 = 7883 if initial values are(1199110 1199080 1205790 1199020) = (049 103 073 minus031) the phase

track converges to a stable equilibrium point If initialvalues are (1199110 1199080 1205790 1199020) = (029 minus079 089 minus115)the phase track is a limit cycle Projections of thecoexisting attractor in two-dimensional plane 119908-120579and three-dimensional space119908-120579-119902 are shown in Fig-ures 5(a) and 5(b) respectively in which the red dotrepresents equilibrium point attractor and the bluelimit cycle represents periodic attractor Within 120590 isin[002745 003255] 119896 isin [minus7685 minus5257] periodicstate always scatters in chaotic region and periodicattractors coexist with chaotic attractors in the dottedregion When 120590 = 003259 119896 = minus5604 if initial val-ues are (1199110 1199080 1205790 1199020) = (minus019 089 minus076 minus141)the phase track converges to a periodic attractorwhile if initial values are (1199110 1199080 1205790 1199020) = (minus106235 minus062 075) the phase track converges to achaotic attractor coexisting attractors are shown inFigures 5(c) and 5(d) the periodic attractor is markedwith red limit cycle and the chaotic attractor ismarked with blue In addition there are other typesof multistability in the system at several combinationsof parameters such as coexistence of different stable

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Figure 5 Projections of coexisting attractors on 119908-120579 plane and 119908-120579-q spaceTable 2 Coexisting attractors at different combinations of parameters

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

120590 = 002468119896 = 9011 (084 minus089 010 minus054)(minus142 049 minus018 minus019) (minus555 minus623 minus8648 minus8697)(minus829 minus901 minus6341 minus6392) 00

Coexistence of stable equilibrium pointwith periodic attractor

120590 = 002603119896 = 5901 (032 minus131 minus043 034)(054 183 minus226 086) (minus1188 minus1212 minus2846 minus2909)(0014 minus1247 minus1783 minus5046) 01

Coexistence of different 1-periodic cycles 120590 = 002805119896 = 6961 (054 183 minus226 086)(minus012 149 141 142) (minus004 minus1165 minus1686 minus6567)(001 minus1122 minus1595 minus5375) 11

Coexistence of 1-periodic cycle with2-periodic cycle

120590 = 003153119896 = minus7313 (144 minus196 minus020 minus121)(030 minus060 049 074) (002 minus214 minus2187 minus7592)(minus002 minus208 minus2115 minus7601) 11

Coexistence of periodic attractor withchaotic attractor

120590 = 003238119896 = minus6263 (minus109 003 055 110)(minus077 037 minus023 112) (003 minus2116 minus2905 minus2990)(231 minus064 minus8520 minus9403) 1232

equilibrium points and coexistence of multiple limitcycles When 120590 = 003171 119896 = minus71432 as shown inFigures 5(e) and 5(f) a red 1-periodic cycle coexistswith a blue 2-periodic cycle When 120590 = 002805 119896 =6961 as shown in Figures 5(g) and 5(h) two differentkinds of 1-periodic cycles coexist When 120590 = 002468119896 = 9011 as shown in Figures 5(i) and 5(j) twodifferent kinds of stable equilibrium points coexistIf parameters are invariable the coexisting attractorsindicate behaviors of the system sensitive to initialconditions the trajectory of supercavitating vehiclemay probably approach two types of attractors whenits initial depth vertical velocity pitch and pitch rateare taken different values namely motion state ofsupercavitating vehicle is likely to be different

(4) Within 120590 isin [00198 002956] 119896 isin [7883 1779] thegreen stable dots scatter in the blue divergent regionWhen 119896 isin [minus95 minus8578] red periodic dots scatterwithin the blue divergent region In the interwovenregions slight change in parameters can always leadto change in motion state of supercavitating vehicleimproper setting of initial conditions would makesupercavitating vehicle capsize Basins of attraction atstable equilibrium points and periodic dots are not

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

It can be derived from above analysis that various coexist-ing attractors exist in the parameter regions marked withdifferent colors in Figure 3 as shown in Table 2 As longas the coexisting attractors and their types are known eachattractor can be associated with all the initial conditions thatmake its trajectory approach this attractor they constituteclustering region of the attractor which is called the basinsof attraction [14 15] So that the final state of the systemis determined by the basins of attraction where initialconditions are located when the initial conditions are nearthe boundary of the attraction basins slight disturbanceor change in parameters will lead to a completely differentmotion of the system The initial conditions will bring highuncertainty in the final destination of the system as welland complex behaviors [16 17] often appear which arelikely to deviate from original expectation of designer andlead to unpredictable motion and thus pose a great threatto the engineering application Therefore it is essential to

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 6 Basins of attraction (a) 120590 = 002468 119896 = 3423 (b) 120590 = 002603 119896 = 5901 (c) 120590 = 002721 119896 = 7473 (d) 120590 = 002655 119896 = 9369understand such abnormal phenomena [18] thoroughly Forcomprehensive analysis of a complex system complexity ofits attractors and its basins of attraction should be analyzedsimultaneously The study of basins of attraction is of highvalue in engineering application Most engineering problemsinvolve not only analysis of local stability and bifurcationunder minor perturbation but also scope of the attractionbasins of steady solutions that is the area of attraction basinswhere steady solution is of same properties With differentlaunching conditions the larger the area of attraction basinsat steady state and periodic state the stronger the robustnessof supercavitating vehicle

41 Basins of Attraction of Stable Equilibrium Point and LimitCycle Select various combinations of parameters from 120590 isin[002620 002973] 119896 isin [7387 14325] namely the regioninterwoven by stable periodic and divergent states thesections of basins of attraction on 1199110-1205790 plane are shownin Figure 6 and basins of attraction are approximate to aparallelogram when launching depth 1199110 and launching pitch1205790 are selected from green region in the figure the lift ofcavitator and fin is equal to the weight of supercavitatingvehicle which make supercavitating vehicle move stablyunder the parameters When initial values are correspondingto the dots within red region tail of supercavitating vehicle

oscillates periodically moving into and out of the super-cavity alternatively When initial values are correspondingto the dots within blue region too high values of initialconditions would lead to divergence of the system and theincapability of supercavitating vehicle to navigate If anyphase dot within basins of attraction of certain attractor istaken as initial condition the system always converges tosuch attractor In combinations of parameters if cavitationnumber varies within a small range the feedback controlgain of fin deflection angle 119896 decreases gradually it can beproven by calculating the area of corresponding attractionbasins that 1198781198861 gt 1198781198871 gt 1198781198881 gt 1198781198891 1198781198861 1198781198871 1198781198881 and 1198781198891 areareas of attraction basins in Figures 6(a) 6(b) 6(c) and6(d) respectivelyTherefore provided that cavitation numbervaries within a small range the area of attraction basinsdecreases with the feedback control gain of fin deflectionangle 119896 With target to different initial launching conditionsthe lower the value of 119896 the smaller the attraction basinof stable equilibrium point and periodic attractor and theweaker the robustness of the system and hence supercav-itating vehicle is more sensitive to external interferencesand becomes unstable In addition when values of initialconditions are corresponding to the dots close to boundariesof the red region and the green region slight interferencecan lead to the switching between stable motion and periodic

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

Figure 7 Basins of attraction (a) 120590 = 002838 119896 = minus5901 (b) 120590 = 003136 119896 = minus6744 (c) 120590 = 003238 119896 = minus6263 (d) 120590 = 003259119896 = minus5604oscillation of the vehicle In practical engineering applicationrobustness of supercavitating vehicle can be improved byadjusting fin deflection angel of supercavitating vehicle andinitial launching conditions

42 Basins of Attraction of Limit Cycle and Chaotic AttractorSelect a few dots from 120590 isin [002745 003255] 119896 isin [minus76851minus52573] namely the regionwhere periodic state interweaveswith chaotic state sections of their basins of attraction on1199110-1205790 plane are shown in Figure 7 the red region representsthe initial values falling into periodic trajectory and super-cavitating vehicle oscillates periodically The black regionrepresents the initial values which draw supercavitating vehi-cle to chaotic state eventually supercavitating vehicle thusbecomes unstable or even capsizesTheblue region representsthe initial values which make the system divergent InFigure 7 under different combinations of parameters basinsof attraction are almost same in shape butwith different sizesThere is a fractal boundary in which slight change in initialconditions may probably trigger the transition of the systemfrom stable periodic state to chaotic state and often leadto weaker robustness and even capsizing of supercavitatingvehicle In practical engineering application it should be

avoided to choose such variable combinations of parametersso as to improve robustness of the system

43 Basins of Attraction of Stable Equilibrium Points Selectseveral dots from 120590 isin [001980 002956] 119896 isin [7883 17794]namely the region where stable state interweaves with diver-gent region Sections of their basins of attraction on 1199110-1205790plane are shown in Figure 8 If initial values 1199110 and 1205790 arewithin green region supercavity can be formed to enclosevehicle and supercavitating vehicle moves stably When theinitial values are within the blue region rather than the greenregion supercavitating vehicle becomes unstable The biggerthe basins of attraction the stronger the robustness of thesystem It can be seen from Figure 8 that if cavitation numbervaries within small range the feedback control gain of findeflection angle 119896 decreases gradually and the area of attrac-tion basins of stable equilibrium points varies with combina-tions of parameters 1198781198862 gt 1198781198872 gt 1198781198882 gt 1198781198892 where 1198781198862 1198781198872 1198781198882and 1198781198892 are area of attraction basins in Figures 8(a) 8(b)8(c) and 8(d) respectively The robustness decreases in suchorder Therefore within such scope provided that cavitationnumber varies within a small range the lower the value of119896 the bigger the area of attraction basins and the stronger

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 8 Basins of attraction (a) 120590 = 002788 119896 = 8874 (b) 120590 = 002838 119896 = 9369 (c) 120590 = 002754 119896 = 1106 (d) 120590 = 002636 119896 = 1004the robustness supercavitating vehicle is more liable to movestably Figure 8 not only shows the range of initial conditionsfor stable motion of supercavitating vehicle but also reflectsthe relationship between the robustness and the parametersIn practical application parameters with bigger basins ofattraction and strong robustness should be selected to ensurestable motion of supercavitating vehicle

44 Basins of Attraction of Limit Cycle Select several dotsfrom 119896 isin [minus89895 minus85786] namely the region where peri-odic state interweaves with stable state Sections of theirbasins of attraction on 1199110-1205790 plane are shown in Figure 9If initial values 1199110 and 1205790 are within the red region peri-odic oscillation occurs due to collision between the tail ofsupercavitating vehicle and the supercavity from time totime supercavitating vehicle is in periodic oscillation Whenthe initial values are corresponding to the dots within theblue region the system is divergent and supercavitatingvehicle capsizes It can be seen from Figure 9 that providedthat cavitation number varies within small range the areaof attraction basins decreases with value of 119896 graduallyTherefore value of 119896 has significant effect on robustness ofsupercavitating vehicle which becomes weaker with increaseof 119896

45 Basins of Attraction of Other Types It can be found fromanalysis of dynamic distribution diagram that coexistence ofstable equilibrium point with limit cycle and coexistence oflimit cycle with chaotic attractor are universal phenomena ofmultistability In addition there are other types ofmultistabil-ity at individual parameters in the system such as coexistenceof 1-periodic cycle with 2-periodic cycle coexistence of twotypes of 1-periodic cycles and coexistence of two types ofstable equilibrium points For the coexistence of 1-periodiccycle with 2-periodic cycle sections of basins of attraction on1199110-1205790 plane are shown in Figures 10(a) and 10(b) respectivelyThe red region represents the initial values which makesupercavitating vehicle approach the 1-periodic trajectoryeventually The dark blue region represents the initial valueswhich make vehicle approach 2-periodic trajectory eventu-allyThe light blue region represents divergence of the systemTherefore different initial launching conditions may proba-bly result in different periods of motion Basin of attractionfor coexistence of two types of 1-periodic cycles is shown inFigure 10(c) the red region and the dark blue region representthe initial values which make supercavitating vehicle fallinto different periods of oscillation eventually With suchcombinations of parameters supercavitating vehicle maydisplay different periodic microoscillation under different

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

Figure 9 Basins of attraction (a) 120590 = 002434 119896 = minus8783 (b) 120590 = 003040 119896 = minus9193 (c) 120590 = 002805 119896 = minus909 (d) 120590 = 003428119896 = minus8988initial conditions Basin of attraction for coexistence of twotypes of stable equilibrium points is shown in Figure 10(d)when initial values correspond with the red region and thedark blue region the motion of supercavitating vehicle willapproach two different types of stable equilibrium pointsSupercavitating vehicle can navigate stably if initial values arewithin the basin of attractionThe system is divergent and thevehicle capsizes once initial values are beyond the boundaryof the basin of attraction

5 Conclusion

In this paper general characteristics of dynamic behaviorsof supercavitating vehicle are studied with dynamic distri-bution diagram based on a 4-dimensional dynamic modelcoexistence of various attractors is confirmed herein and therelation between the robustness of supercavitating vehicleand the parameters of system and initial values is obtainedthough multistability analysis Following conclusions can bedrawn

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

dynamic distribution diagram the relationshipbetween dynamic behaviors and variable parameterscan be obtained And the regions of parameterswhere coexisting attractors are likely to be found aredisclosed providing the basis for setting parametersfor stable motion

(2) Basins of attraction vary with parameters of thesystem some attractors will die and new attractorswill be generated Basins of attraction can be usedto determine the range of initial conditions for sta-ble motion of supercavitating vehicle and unstablemotion can be prevented by adjusting initial values oflaunch

(3) Generally supercavitating vehicle has three stablestates including stable periodic and chaotic statesUnder appropriate combinations of parameters thereare various motions of multistability such as coex-istence of two types of stable equilibrium pointscoexistence of a stable equilibrium point with alimit cycle coexistence of a limit cycle with chaoticattractor and coexistence of multiple limit cycles

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Figure 10 Basins of attraction (a) 120590 = 003171 119896 = minus7143 (b) 120590 = 003187 119896 = minus7094 (c) 120590 = 002805 119896 = 6961 (d) 120590 = 002464119896 = 9011When parameters are fixed supercavitating vehiclemay display different states of motion under differentinitial values

(4) Provided that cavitation number varies within a smallrange robustness of the system becomes weaker withthe increase of feedback control gain of fin deflectionangle 119896 size of basins of attraction becomes smallerand robustness of the system becomes weaker Sys-tematic parameters with greater basins of attractioncan be selected to lessen sensitivity to external inter-ference and improve robustness of supercavitatingvehicle

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant nos 11402116 and 11472163) and

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

[1] A Ducoin B Huang and Y L Young ldquoNumerical modeling ofunsteady cavitating flows around a stationary hydrofoilrdquo Inter-national Journal of Rotating Machinery vol 2012 Article ID215678 17 pages 2012

[2] B Vanek J Bokor G J Balas and R E A Arndt ldquoLongitudinalmotion control of a high-speed supercavitation vehiclerdquo Journalof Vibration and Control vol 13 no 2 pp 159ndash184 2007

[3] Q-T Li Y-S He L-P Xue and Y-Q Yang ldquoA numericalsimulation of pitching motion of the ventilated supercavitingvehicle around its noserdquo Chinese Journal of Hydrodynamics Avol 26 no 6 pp 689ndash696 2011

[4] J Dzielski and A Kurdila ldquoA benchmark control problem forsupercavitating vehicles and an initial investigation of solu-tionsrdquo Journal of Vibration andControl vol 9 no 7 pp 791ndash8042003

[5] M A Hassouneh V Nguyen B Balachandran and E H AbedldquoStability analysis and control of supercavitating vehicles with

Advances in Mathematical Physics 13

advection delayrdquo Journal of Computational and Nonlinear Dy-namics vol 8 no 2 Article ID 021003 10 pages 2013

[6] B Vanek J Bokor and G Balas ldquoTheoretical aspects of high-speed supercavitation vehicle controlrdquo American Control Con-ference vol 6 no 3 pp 1ndash4 2006

[7] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007

[8] G Lin B Balachandran and E H Abed ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASME Inter-national Mechanical Engineering Congress and Exposition pp293ndash300 Chicago Ill USA November 2006

[9] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008

[10] B Vanek andG Balas ldquoControl of high-speed underwater vehi-clesrdquo Control of Uncertain Systems Modeling Approximationand Design no 329 pp 25ndash44 2006

[11] A Goel Robust control of supercavitating vehicles in the presenceof dynamic and uncertain cavity [Doctorrsquos thesis] University ofFlorida Gainesville Fla USA 2005

[12] X H Zhao Y Sun Z K Qi and M Y Han ldquoCatastrophecharacteristics and control of pitching supercavitating vehiclesat fixed depthsrdquo Ocean Engineering vol 112 pp 185ndash194 2016

[13] M L Wang and G L Zhao ldquoRobust controller design forsupercavitating vehicles based on BTT maneuvering strategyrdquoin Proceedings of the International Conference on Mechatronicsamp Automation pp 227ndash231 2007

[14] G A Leonov and N V Kuznetsov ldquoHidden attractors in dy-namical systems from hidden oscillations in hilbert-kol-mogorov Aizerman and Kalman problems to hidden chaoticattractor in chua circuitsrdquo International Journal of Bifurcationand Chaos vol 23 no 1 Article ID 1330002 2013

[15] M Chen M Li Q Yu B Bao Q Xu and J Wang ldquoDynamicsof self-excited attractors and hidden attractors in generalizedmemristor-based Chuarsquos circuitrdquo Nonlinear Dynamics vol 81no 1-2 pp 215ndash226 2015

[16] J C A de Bruin A Doris N van de Wouw W P M HHeemels and H Nijmeijer ldquoControl of mechanical motionsystems with non-collocation of actuation and friction a Popovcriterion approach for input-to-state stability and set-valuednonlinearitiesrdquo Automatica vol 45 no 2 pp 405ndash415 2009

[17] M A Kiseleva N V Kuznetsov G A Leonov and P Neit-taanmaki ldquoDrilling systems failures and hidden oscillationsrdquoin Proceedings of the IEEE 4th International Conference onNonlinear Science and Complexity (NSC rsquo12) pp 109ndash112 IEEEBudapest Hungary August 2012

[18] G A Leonov N V Kuznetsov O A Kuznetsova SM Seledzhiand V I Vagaitsev ldquoHidden oscillations in dynamical systemsrdquoWSEAS Transactions on Systems and Control vol 6 no 2 pp54ndash67 2011

[19] T Lauvdal R M Murray and T I Fossen ldquoStabilizationof integrator chains in the presence of magnitude and ratesaturations a gain scheduling approachrdquo in Proceedings of the36th IEEE Conference on Decision and Control pp 4004ndash4005December 1997

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

In the above equation 119862119909 denotes the cavitator dragcoefficient and119862119909 = 1198621199090(1+120590) where1198621199090 = 082 120572119888 denotesthe angle of attack due to cavitator deflection 120575119888 and verticalvelocity 119881 Similarly the lift on the fin is approximately [4]119865fin = minus1198991212058712058811987721198991198812119862119909120572119891 (2)

In (2) the parameter 119899 represents the efficiency of the finwhich is the ratio of the length of the fin immersed in thewater to the total length of the fin and 120572119891 denotes the angle ofattack due to fin deflection 120575119891 and vertical velocity 119881

When the vehicle body navigates in supercavity due tothe change of relative position of the body and the cavitythe tail and the cavity will touch with each other which willproduce a complex nonlinear planning force resulting invibration and shock the normalized force 119865planing is [4]

119865planing = minus1198812 [1 minus ( 1198771015840ℎ119877 + 1198771015840)2]( 1 + ℎ1 + 2ℎ)120572 (3)

In the previous equation 1198771015840 = 119877119888 minus 119877 119877119888 denotes theradius of the cavity and 119877 denotes radius of vehicle Theimmersion depth ℎ and the angle of attack 120572 in the planningforce are given by [4]

ℎ = 0 |119908| lt 119908th = (119877119888 minus 119877)119881119871119871 |119908|119877119881 minus 119877119888 minus 119877119877 otherwise

120572 = 119908 minus 119888119881 119908119881 gt 0119908 + 119888119881 otherwise

(4)

In (4)119908 is the vertical velocity119871 is the vehicle length and119888 is the cavity radius contraction rateDzielski and Kurdila [4] presented a simplified 4-dimen-

sional dynamic model of supercavitating vehicle Althoughonly the effect of angle of attack on planning force isconsidered in the model motion characteristics of supercav-itating vehicle in vertical plane can be described qualitativelyFollowing the work of Dzielski and Kurdila in present worka dynamic model of supercavitating vehicle is established bytwo bifurcation parameters

The center of top surface of the disk-shaped cavitatoron head of supercavitating vehicle is taken as the originof coordinate system The four state variables are used todescribe dynamic of supercavitating vehicle in the modelnamely 119911 119908 120579 and 119902 wherein 119911 represents the depth wherethe body is located 119908 is the vertical velocity and 120579 and 119902refer to the pitch and pitch rate respectively The verticalvelocity119908 is perpendicular to the axial line of supercavitatingvehicle and forward velocity 119881 is parallel to the axial lineIn addition the system has two control inputs namelycavitator deflection angle 120575119888 and fin deflection angle 120575119890 Inthe classic control law presented by Dzielski and Kurdila [4]120575119890 = 0 and 120575119888 = 15119911-30120579-03119902 However supercavitating

vehicle would lack supportive force of fin due to 120575119890 = 0If the lift on the cavitator 119865cavitator could not overcome theweight of supercavitating vehicle vehicle would immerseinto supercavity result in unstable motion Therefore thecontrol law in the paper is chosen as 120575119890 = minus119896119911 1205751198881 =15119911-30120579-03119902 in which 119896 refers to the feedback control gainof 119911 According to fluid dynamics exerted on different partsof supercavitating vehicle the dynamic model [7] can beestablished with cavitation number 120590 and feedback controlgain of fin deflection angle 119896 as variable parameters

(120579119902) = (0 1 minus119881 00 11988622 0 119886240 0 0 10 11988642 0 11988644)(119911119908120579119902)

+( 0 011988721 119887220 011988741 11988742)(120575119890120575119888) +(0119888200)

+( 0119889201198894)119865planing

(5)

In (5)

11988622 = 119862119881119879119898 (minus1 minus 119899119871 ) 119878 + 173611989911987111988624 = 119881119879119878 (119862minus119899119898 + 79) minus 119881119879(119862minus119899119898 + 1736) 1736119871211988642 = 119862119881119879119898 (1736 minus 1111989936 ) 11988644 = minus1111986211988111987911989911987136119898 11988721 = 1198621198812119879119899119898 (minus119878119871 + 1711987136 ) 11988722 = minus1198621198812119879119878119898119871 11988741 = minus11119862119881211987911989936119898 11988742 = 17119862119881211987936119898 1198882 = 1198921198892 = 119879119898 (minus1711987136 + 119878119871)

4 Advances in Mathematical Physics

Table 1 Parameters of supercavitating vehicle model

Parameter Description Value119892 Acceleration of gravity 981ms2

m Density ratio (120588119898120588) 2n Tail efficiency 05119877119899 Radius of cavitator 00191mR Radius of vehicle 00508mL Length of vehicle 18m120590 Cavitation number [001980 003680]1198621199090 Coefficient of lift 082

0024 0028 0032 00360020

minus80

minus60

minus40

minus20

0

20

40

120590

k

Figure 3 Distribution diagram of dynamic behaviors of supercavi-tating vehicle

1198894 = 1111987936119898119878 = 11601198772 + 1331198712405 119879 = 171198789 minus 28911987121296 119862 = 051198621199090 (1 + 120590) (119877119899119877 )2

(6)119898 (120588119898120588) denotes density ratio where 120588119898 is the specificationof a uniform density for the vehicle and 120588 is the density ofwater

3 General Dynamic Characteristics ofSupercavitating Vehicle

Values of parameters for supercavitating vehicle model aregiven in Table 1 [7] Based on dynamic model (5) the initialconditions are selected randomly According to the Lyapunovstability theory the stable solutions periodic solutions andchaotic solutions of the model are represented by greenred and black in Figure 3 respectively The distributiondiagram of dynamic behaviors defined by cavitation number120590 and feedback control gain of fin deflection angle 119896 as

the bifurcation parameters is drawn And the dependenceof dynamic behaviors on 120590 and 119896 is described As shownin Figure 3 if value of 120590 119896 is within the green regionthe max Lyapunov exponent of (5) is a negative and statevariables 119911 119908 120579 and 119902 converge on stable equilibriumpoints supercavitating vehicle can move steadily If the point(120590 119896) is within the red region the max Lyapunov exponentof (5) is zero and state variables 119911 119908 120579 and 119902 oscillateperiodically centering on stable equilibriumpoints thereforesupercavitating vehicle shock periodically If (120590 119896) is withinthe black region the max Lyapunov exponent of (5) is apositive and state variables 119911 119908 120579 and 119902 oscillate violentlyand irregularly violent vibrations and shocks will occur andthen supercavitating vehicle will capsizeThe light blue regionrepresents where the system is divergent and vehicle cannotnavigate

Figure 3 reflects different dynamics of supercavitatingvehicle completely when the parameters 120590 and 119896 changesimultaneously The range of parameters corresponding tostable motion of vehicle can be determined by the dynamicdistribution diagramWhen cavitation number 120590 is constantthe value of feedback control gain of fin deflection angle 119896can be adjusted within the green stable region and the redperiodic region to realize stable motion of supercavitatingvehicle efficiently which is instructive to stability control ofsupercavitating vehicle It can be seen from Figure 3 that onehas the following

(1) The horizontal section is bifurcation diagram of thesystem varying with 120590 while the vertical section isbifurcation diagram of the system varying with 119896Thebifurcation diagram reflects the rules of changes ofthe system with parameters and complex nonlinearphysical phenomena generatedThe Hopf bifurcationalways occurs when the system switches from steadystate to periodic state so the boundary betweenthe red region and the green region in Figure 3 iscritical line between steady state and periodic statealso called the Hopf bifurcation line The bound-ary between the red region and the black regionrepresents the switching between the periodic stateand the chaotic state There are nonlinear physicalphenomena such as the cutting bifurcation or period-doubling bifurcation at this boundary

(2) Supercavitating vehicle has three stable states includ-ing stable motion periodic motion and chaotic stateSelect one point from the three regions respectivelythe projections of phase tracks on 119908-120579 plane areshown in Figure 4 Select 120590 = 002249 119896 = minus1405from the green region as shown in Figure 4(a)vertical velocity119908 and pitch 120579 are attracted to a stableequilibrium point functioned by feedback controllaw supercavitating vehicle is exerted by forces inequilibrium state navigating stably in fixed positionand attitude inside supercavity with oblique smallangle of attack Select 120590 = 003459 119896 = minus175 ran-domly from the red region as shown in Figure 4(b)mapping of the system forms a closed limit cycle andthe limit cycle intersects the red switching critical line

Advances in Mathematical Physics 5

minus001

0

001

015 020010

120579(rad)

w (mmiddotsminus1)

(a)

minus001

0

001

002

0 5minus5

120579(rad)

w (mmiddotsminus1)

(b)

minus0004

minus0002

0

0002

0004

minus1 0 1 2minus2

120579(rad)

w (mmiddotsminus1)

(c)

Figure 4 Phase tracks of three stable states on 119908-120579 plane (a) 120590 = 002249 119896 = minus1405 (b) 120590 = 003459 119896 = minus175 (c) 120590 = 003040119896 = minus5913119908 = 119908th where 119908th = (119877119888 minus 119877)119881119871 [7] 119877119888 is theradius of supercavity and vertical velocity 119908 fluc-tuates around 119908th Tail of supercavitating vehicleoscillates from time to time touching the supercavitywhich leads to periodic change of planning forcesometimes the tail penetrates the supercavity andinserts into water to generate planning force andsometimes it is enclosed by the supercavity and thusno planning force is generated and supercavitatingvehicle oscillates periodically Select parameters 120590 =003040 119896 = minus5913 randomly from the red regionappearance of chaotic attractor indicates that super-cavitating vehicle has complex nonlinear dynamicbehavior and is likely to capsize In practical applica-tion of engineering effective control should be takento prevent occurrence of such circumstance

(3) Within 120590 isin [00262 002973] 119896 isin [7387 1432] thegreen stable region the red periodic region and thedivergent region are interwoven stable equilibriumpoints coexist with periodic attractors in the regionWhen 120590 = 002771 119896 = 7883 if initial values are(1199110 1199080 1205790 1199020) = (049 103 073 minus031) the phase

track converges to a stable equilibrium point If initialvalues are (1199110 1199080 1205790 1199020) = (029 minus079 089 minus115)the phase track is a limit cycle Projections of thecoexisting attractor in two-dimensional plane 119908-120579and three-dimensional space119908-120579-119902 are shown in Fig-ures 5(a) and 5(b) respectively in which the red dotrepresents equilibrium point attractor and the bluelimit cycle represents periodic attractor Within 120590 isin[002745 003255] 119896 isin [minus7685 minus5257] periodicstate always scatters in chaotic region and periodicattractors coexist with chaotic attractors in the dottedregion When 120590 = 003259 119896 = minus5604 if initial val-ues are (1199110 1199080 1205790 1199020) = (minus019 089 minus076 minus141)the phase track converges to a periodic attractorwhile if initial values are (1199110 1199080 1205790 1199020) = (minus106235 minus062 075) the phase track converges to achaotic attractor coexisting attractors are shown inFigures 5(c) and 5(d) the periodic attractor is markedwith red limit cycle and the chaotic attractor ismarked with blue In addition there are other typesof multistability in the system at several combinationsof parameters such as coexistence of different stable

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Figure 5 Projections of coexisting attractors on 119908-120579 plane and 119908-120579-q spaceTable 2 Coexisting attractors at different combinations of parameters

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

120590 = 002468119896 = 9011 (084 minus089 010 minus054)(minus142 049 minus018 minus019) (minus555 minus623 minus8648 minus8697)(minus829 minus901 minus6341 minus6392) 00

Coexistence of stable equilibrium pointwith periodic attractor

120590 = 002603119896 = 5901 (032 minus131 minus043 034)(054 183 minus226 086) (minus1188 minus1212 minus2846 minus2909)(0014 minus1247 minus1783 minus5046) 01

Coexistence of different 1-periodic cycles 120590 = 002805119896 = 6961 (054 183 minus226 086)(minus012 149 141 142) (minus004 minus1165 minus1686 minus6567)(001 minus1122 minus1595 minus5375) 11

Coexistence of 1-periodic cycle with2-periodic cycle

120590 = 003153119896 = minus7313 (144 minus196 minus020 minus121)(030 minus060 049 074) (002 minus214 minus2187 minus7592)(minus002 minus208 minus2115 minus7601) 11

Coexistence of periodic attractor withchaotic attractor

120590 = 003238119896 = minus6263 (minus109 003 055 110)(minus077 037 minus023 112) (003 minus2116 minus2905 minus2990)(231 minus064 minus8520 minus9403) 1232

equilibrium points and coexistence of multiple limitcycles When 120590 = 003171 119896 = minus71432 as shown inFigures 5(e) and 5(f) a red 1-periodic cycle coexistswith a blue 2-periodic cycle When 120590 = 002805 119896 =6961 as shown in Figures 5(g) and 5(h) two differentkinds of 1-periodic cycles coexist When 120590 = 002468119896 = 9011 as shown in Figures 5(i) and 5(j) twodifferent kinds of stable equilibrium points coexistIf parameters are invariable the coexisting attractorsindicate behaviors of the system sensitive to initialconditions the trajectory of supercavitating vehiclemay probably approach two types of attractors whenits initial depth vertical velocity pitch and pitch rateare taken different values namely motion state ofsupercavitating vehicle is likely to be different

(4) Within 120590 isin [00198 002956] 119896 isin [7883 1779] thegreen stable dots scatter in the blue divergent regionWhen 119896 isin [minus95 minus8578] red periodic dots scatterwithin the blue divergent region In the interwovenregions slight change in parameters can always leadto change in motion state of supercavitating vehicleimproper setting of initial conditions would makesupercavitating vehicle capsize Basins of attraction atstable equilibrium points and periodic dots are not

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

It can be derived from above analysis that various coexist-ing attractors exist in the parameter regions marked withdifferent colors in Figure 3 as shown in Table 2 As longas the coexisting attractors and their types are known eachattractor can be associated with all the initial conditions thatmake its trajectory approach this attractor they constituteclustering region of the attractor which is called the basinsof attraction [14 15] So that the final state of the systemis determined by the basins of attraction where initialconditions are located when the initial conditions are nearthe boundary of the attraction basins slight disturbanceor change in parameters will lead to a completely differentmotion of the system The initial conditions will bring highuncertainty in the final destination of the system as welland complex behaviors [16 17] often appear which arelikely to deviate from original expectation of designer andlead to unpredictable motion and thus pose a great threatto the engineering application Therefore it is essential to

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 6 Basins of attraction (a) 120590 = 002468 119896 = 3423 (b) 120590 = 002603 119896 = 5901 (c) 120590 = 002721 119896 = 7473 (d) 120590 = 002655 119896 = 9369understand such abnormal phenomena [18] thoroughly Forcomprehensive analysis of a complex system complexity ofits attractors and its basins of attraction should be analyzedsimultaneously The study of basins of attraction is of highvalue in engineering application Most engineering problemsinvolve not only analysis of local stability and bifurcationunder minor perturbation but also scope of the attractionbasins of steady solutions that is the area of attraction basinswhere steady solution is of same properties With differentlaunching conditions the larger the area of attraction basinsat steady state and periodic state the stronger the robustnessof supercavitating vehicle

41 Basins of Attraction of Stable Equilibrium Point and LimitCycle Select various combinations of parameters from 120590 isin[002620 002973] 119896 isin [7387 14325] namely the regioninterwoven by stable periodic and divergent states thesections of basins of attraction on 1199110-1205790 plane are shownin Figure 6 and basins of attraction are approximate to aparallelogram when launching depth 1199110 and launching pitch1205790 are selected from green region in the figure the lift ofcavitator and fin is equal to the weight of supercavitatingvehicle which make supercavitating vehicle move stablyunder the parameters When initial values are correspondingto the dots within red region tail of supercavitating vehicle

oscillates periodically moving into and out of the super-cavity alternatively When initial values are correspondingto the dots within blue region too high values of initialconditions would lead to divergence of the system and theincapability of supercavitating vehicle to navigate If anyphase dot within basins of attraction of certain attractor istaken as initial condition the system always converges tosuch attractor In combinations of parameters if cavitationnumber varies within a small range the feedback controlgain of fin deflection angle 119896 decreases gradually it can beproven by calculating the area of corresponding attractionbasins that 1198781198861 gt 1198781198871 gt 1198781198881 gt 1198781198891 1198781198861 1198781198871 1198781198881 and 1198781198891 areareas of attraction basins in Figures 6(a) 6(b) 6(c) and6(d) respectivelyTherefore provided that cavitation numbervaries within a small range the area of attraction basinsdecreases with the feedback control gain of fin deflectionangle 119896 With target to different initial launching conditionsthe lower the value of 119896 the smaller the attraction basinof stable equilibrium point and periodic attractor and theweaker the robustness of the system and hence supercav-itating vehicle is more sensitive to external interferencesand becomes unstable In addition when values of initialconditions are corresponding to the dots close to boundariesof the red region and the green region slight interferencecan lead to the switching between stable motion and periodic

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

Figure 7 Basins of attraction (a) 120590 = 002838 119896 = minus5901 (b) 120590 = 003136 119896 = minus6744 (c) 120590 = 003238 119896 = minus6263 (d) 120590 = 003259119896 = minus5604oscillation of the vehicle In practical engineering applicationrobustness of supercavitating vehicle can be improved byadjusting fin deflection angel of supercavitating vehicle andinitial launching conditions

42 Basins of Attraction of Limit Cycle and Chaotic AttractorSelect a few dots from 120590 isin [002745 003255] 119896 isin [minus76851minus52573] namely the regionwhere periodic state interweaveswith chaotic state sections of their basins of attraction on1199110-1205790 plane are shown in Figure 7 the red region representsthe initial values falling into periodic trajectory and super-cavitating vehicle oscillates periodically The black regionrepresents the initial values which draw supercavitating vehi-cle to chaotic state eventually supercavitating vehicle thusbecomes unstable or even capsizesTheblue region representsthe initial values which make the system divergent InFigure 7 under different combinations of parameters basinsof attraction are almost same in shape butwith different sizesThere is a fractal boundary in which slight change in initialconditions may probably trigger the transition of the systemfrom stable periodic state to chaotic state and often leadto weaker robustness and even capsizing of supercavitatingvehicle In practical engineering application it should be

avoided to choose such variable combinations of parametersso as to improve robustness of the system

43 Basins of Attraction of Stable Equilibrium Points Selectseveral dots from 120590 isin [001980 002956] 119896 isin [7883 17794]namely the region where stable state interweaves with diver-gent region Sections of their basins of attraction on 1199110-1205790plane are shown in Figure 8 If initial values 1199110 and 1205790 arewithin green region supercavity can be formed to enclosevehicle and supercavitating vehicle moves stably When theinitial values are within the blue region rather than the greenregion supercavitating vehicle becomes unstable The biggerthe basins of attraction the stronger the robustness of thesystem It can be seen from Figure 8 that if cavitation numbervaries within small range the feedback control gain of findeflection angle 119896 decreases gradually and the area of attrac-tion basins of stable equilibrium points varies with combina-tions of parameters 1198781198862 gt 1198781198872 gt 1198781198882 gt 1198781198892 where 1198781198862 1198781198872 1198781198882and 1198781198892 are area of attraction basins in Figures 8(a) 8(b)8(c) and 8(d) respectively The robustness decreases in suchorder Therefore within such scope provided that cavitationnumber varies within a small range the lower the value of119896 the bigger the area of attraction basins and the stronger

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Figure 8 Basins of attraction (a) 120590 = 002788 119896 = 8874 (b) 120590 = 002838 119896 = 9369 (c) 120590 = 002754 119896 = 1106 (d) 120590 = 002636 119896 = 1004the robustness supercavitating vehicle is more liable to movestably Figure 8 not only shows the range of initial conditionsfor stable motion of supercavitating vehicle but also reflectsthe relationship between the robustness and the parametersIn practical application parameters with bigger basins ofattraction and strong robustness should be selected to ensurestable motion of supercavitating vehicle

44 Basins of Attraction of Limit Cycle Select several dotsfrom 119896 isin [minus89895 minus85786] namely the region where peri-odic state interweaves with stable state Sections of theirbasins of attraction on 1199110-1205790 plane are shown in Figure 9If initial values 1199110 and 1205790 are within the red region peri-odic oscillation occurs due to collision between the tail ofsupercavitating vehicle and the supercavity from time totime supercavitating vehicle is in periodic oscillation Whenthe initial values are corresponding to the dots within theblue region the system is divergent and supercavitatingvehicle capsizes It can be seen from Figure 9 that providedthat cavitation number varies within small range the areaof attraction basins decreases with value of 119896 graduallyTherefore value of 119896 has significant effect on robustness ofsupercavitating vehicle which becomes weaker with increaseof 119896

45 Basins of Attraction of Other Types It can be found fromanalysis of dynamic distribution diagram that coexistence ofstable equilibrium point with limit cycle and coexistence oflimit cycle with chaotic attractor are universal phenomena ofmultistability In addition there are other types ofmultistabil-ity at individual parameters in the system such as coexistenceof 1-periodic cycle with 2-periodic cycle coexistence of twotypes of 1-periodic cycles and coexistence of two types ofstable equilibrium points For the coexistence of 1-periodiccycle with 2-periodic cycle sections of basins of attraction on1199110-1205790 plane are shown in Figures 10(a) and 10(b) respectivelyThe red region represents the initial values which makesupercavitating vehicle approach the 1-periodic trajectoryeventually The dark blue region represents the initial valueswhich make vehicle approach 2-periodic trajectory eventu-allyThe light blue region represents divergence of the systemTherefore different initial launching conditions may proba-bly result in different periods of motion Basin of attractionfor coexistence of two types of 1-periodic cycles is shown inFigure 10(c) the red region and the dark blue region representthe initial values which make supercavitating vehicle fallinto different periods of oscillation eventually With suchcombinations of parameters supercavitating vehicle maydisplay different periodic microoscillation under different

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

Figure 9 Basins of attraction (a) 120590 = 002434 119896 = minus8783 (b) 120590 = 003040 119896 = minus9193 (c) 120590 = 002805 119896 = minus909 (d) 120590 = 003428119896 = minus8988initial conditions Basin of attraction for coexistence of twotypes of stable equilibrium points is shown in Figure 10(d)when initial values correspond with the red region and thedark blue region the motion of supercavitating vehicle willapproach two different types of stable equilibrium pointsSupercavitating vehicle can navigate stably if initial values arewithin the basin of attractionThe system is divergent and thevehicle capsizes once initial values are beyond the boundaryof the basin of attraction

5 Conclusion

In this paper general characteristics of dynamic behaviorsof supercavitating vehicle are studied with dynamic distri-bution diagram based on a 4-dimensional dynamic modelcoexistence of various attractors is confirmed herein and therelation between the robustness of supercavitating vehicleand the parameters of system and initial values is obtainedthough multistability analysis Following conclusions can bedrawn

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

dynamic distribution diagram the relationshipbetween dynamic behaviors and variable parameterscan be obtained And the regions of parameterswhere coexisting attractors are likely to be found aredisclosed providing the basis for setting parametersfor stable motion

(2) Basins of attraction vary with parameters of thesystem some attractors will die and new attractorswill be generated Basins of attraction can be usedto determine the range of initial conditions for sta-ble motion of supercavitating vehicle and unstablemotion can be prevented by adjusting initial values oflaunch

(3) Generally supercavitating vehicle has three stablestates including stable periodic and chaotic statesUnder appropriate combinations of parameters thereare various motions of multistability such as coex-istence of two types of stable equilibrium pointscoexistence of a stable equilibrium point with alimit cycle coexistence of a limit cycle with chaoticattractor and coexistence of multiple limit cycles

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Figure 10 Basins of attraction (a) 120590 = 003171 119896 = minus7143 (b) 120590 = 003187 119896 = minus7094 (c) 120590 = 002805 119896 = 6961 (d) 120590 = 002464119896 = 9011When parameters are fixed supercavitating vehiclemay display different states of motion under differentinitial values

(4) Provided that cavitation number varies within a smallrange robustness of the system becomes weaker withthe increase of feedback control gain of fin deflectionangle 119896 size of basins of attraction becomes smallerand robustness of the system becomes weaker Sys-tematic parameters with greater basins of attractioncan be selected to lessen sensitivity to external inter-ference and improve robustness of supercavitatingvehicle

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant nos 11402116 and 11472163) and

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

[1] A Ducoin B Huang and Y L Young ldquoNumerical modeling ofunsteady cavitating flows around a stationary hydrofoilrdquo Inter-national Journal of Rotating Machinery vol 2012 Article ID215678 17 pages 2012

[2] B Vanek J Bokor G J Balas and R E A Arndt ldquoLongitudinalmotion control of a high-speed supercavitation vehiclerdquo Journalof Vibration and Control vol 13 no 2 pp 159ndash184 2007

[3] Q-T Li Y-S He L-P Xue and Y-Q Yang ldquoA numericalsimulation of pitching motion of the ventilated supercavitingvehicle around its noserdquo Chinese Journal of Hydrodynamics Avol 26 no 6 pp 689ndash696 2011

[4] J Dzielski and A Kurdila ldquoA benchmark control problem forsupercavitating vehicles and an initial investigation of solu-tionsrdquo Journal of Vibration andControl vol 9 no 7 pp 791ndash8042003

[5] M A Hassouneh V Nguyen B Balachandran and E H AbedldquoStability analysis and control of supercavitating vehicles with

Advances in Mathematical Physics 13

advection delayrdquo Journal of Computational and Nonlinear Dy-namics vol 8 no 2 Article ID 021003 10 pages 2013

[6] B Vanek J Bokor and G Balas ldquoTheoretical aspects of high-speed supercavitation vehicle controlrdquo American Control Con-ference vol 6 no 3 pp 1ndash4 2006

[7] G Lin B Balachandran and E H Abed ldquoNonlinear dynamicsand bifurcations of a supercavitating vehiclerdquo IEEE Journal ofOceanic Engineering vol 32 no 4 pp 753ndash761 2007

[8] G Lin B Balachandran and E H Abed ldquoBifurcation behaviorof a supercavitating vehiclerdquo in Proceedings of the ASME Inter-national Mechanical Engineering Congress and Exposition pp293ndash300 Chicago Ill USA November 2006

[9] G J Lin B Balachandran and E H Abed ldquoDynamics andcontrol of supercavitating vehiclesrdquo Journal of Dynamic SystemsMeasurement and Control vol 130 no 2 Article ID 021003 pp281ndash287 2008

[10] B Vanek andG Balas ldquoControl of high-speed underwater vehi-clesrdquo Control of Uncertain Systems Modeling Approximationand Design no 329 pp 25ndash44 2006

[11] A Goel Robust control of supercavitating vehicles in the presenceof dynamic and uncertain cavity [Doctorrsquos thesis] University ofFlorida Gainesville Fla USA 2005

[12] X H Zhao Y Sun Z K Qi and M Y Han ldquoCatastrophecharacteristics and control of pitching supercavitating vehiclesat fixed depthsrdquo Ocean Engineering vol 112 pp 185ndash194 2016

[13] M L Wang and G L Zhao ldquoRobust controller design forsupercavitating vehicles based on BTT maneuvering strategyrdquoin Proceedings of the International Conference on Mechatronicsamp Automation pp 227ndash231 2007

[14] G A Leonov and N V Kuznetsov ldquoHidden attractors in dy-namical systems from hidden oscillations in hilbert-kol-mogorov Aizerman and Kalman problems to hidden chaoticattractor in chua circuitsrdquo International Journal of Bifurcationand Chaos vol 23 no 1 Article ID 1330002 2013

[15] M Chen M Li Q Yu B Bao Q Xu and J Wang ldquoDynamicsof self-excited attractors and hidden attractors in generalizedmemristor-based Chuarsquos circuitrdquo Nonlinear Dynamics vol 81no 1-2 pp 215ndash226 2015

[16] J C A de Bruin A Doris N van de Wouw W P M HHeemels and H Nijmeijer ldquoControl of mechanical motionsystems with non-collocation of actuation and friction a Popovcriterion approach for input-to-state stability and set-valuednonlinearitiesrdquo Automatica vol 45 no 2 pp 405ndash415 2009

[17] M A Kiseleva N V Kuznetsov G A Leonov and P Neit-taanmaki ldquoDrilling systems failures and hidden oscillationsrdquoin Proceedings of the IEEE 4th International Conference onNonlinear Science and Complexity (NSC rsquo12) pp 109ndash112 IEEEBudapest Hungary August 2012

[18] G A Leonov N V Kuznetsov O A Kuznetsova SM Seledzhiand V I Vagaitsev ldquoHidden oscillations in dynamical systemsrdquoWSEAS Transactions on Systems and Control vol 6 no 2 pp54ndash67 2011

[19] T Lauvdal R M Murray and T I Fossen ldquoStabilizationof integrator chains in the presence of magnitude and ratesaturations a gain scheduling approachrdquo in Proceedings of the36th IEEE Conference on Decision and Control pp 4004ndash4005December 1997

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4 Advances in Mathematical Physics

Table 1 Parameters of supercavitating vehicle model

Parameter Description Value119892 Acceleration of gravity 981ms2

0024 0028 0032 00360020

minus80

minus60

minus40

minus20

0

20

40

120590

k

Figure 3 Distribution diagram of dynamic behaviors of supercavi-tating vehicle

3 General Dynamic Characteristics ofSupercavitating Vehicle

Advances in Mathematical Physics 5

minus001

0

001

015 020010

120579(rad)

w (mmiddotsminus1)

(a)

minus001

0

001

002

0 5minus5

120579(rad)

w (mmiddotsminus1)

(b)

minus0004

minus0002

0

0002

0004

minus1 0 1 2minus2

120579(rad)

w (mmiddotsminus1)

(c)

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

Coexistence of stable equilibrium pointwith periodic attractor

Coexistence of 1-periodic cycle with2-periodic cycle

Coexistence of periodic attractor withchaotic attractor

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

minus001

0

001

015 020010

120579(rad)

w (mmiddotsminus1)

(a)

minus001

0

001

002

0 5minus5

120579(rad)

w (mmiddotsminus1)

(b)

minus0004

minus0002

0

0002

0004

minus1 0 1 2minus2

120579(rad)

w (mmiddotsminus1)

(c)

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

Coexistence of stable equilibrium pointwith periodic attractor

Coexistence of 1-periodic cycle with2-periodic cycle

Coexistence of periodic attractor withchaotic attractor

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

minus4 minus2 0 2 4 6minus6

002

001

0

minus001

minus002

120579(rad)

w (mmiddotsminus1)

(a)

minus50

5

minus002minus001

0001

002

w (mmiddotsminus1 )

120579 (rad)

minus4

minus2

0

2

4

q(radmiddotsminus

1)

(b)

minus001

0

001

120579(rad)

minus2 0 2 4minus4w (mmiddotsminus1)

(c)

minus20

2

minus001

0

001minus2

minus1

0

1

2

q(radmiddotsminus

1)

w (mmiddotsminus1 )

120579 (rad)

(d)

minus0008

minus0004

0

0004

0008

120579(rad)

minus1 0 1 2minus2w (mmiddotsminus1)

(e)

minus2minus1

01 2

minus0008minus00040

00040008

w (mmiddotsminus1 )

120579 (rad)

minus1

0

1

q(radmiddotsminus

1)

(f)

minus2 0 2 4minus4w (mmiddotsminus1)

minus004

minus002

0

002

004

120579(rad)

(g)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )

120579 (rad)

minus05

0

05

q(radmiddotsminus

1)

(h)

Figure 5 Continued

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

Coexistence of stable equilibrium pointwith periodic attractor

Coexistence of 1-periodic cycle with2-periodic cycle

Coexistence of periodic attractor withchaotic attractor

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

minus008

minus004

0

004

008120579(rad)

minus4 0 4 8minus8w (mmiddotsminus1)

(i)

minus4minus2

02

4

minus004minus002

0002

004

w (mmiddotsminus1 )120579 (rad)

minus01

0

01

q(radmiddotsminus

1)

(j)

Type Parameters Initial conditions Lyapunov exponent Attractordimensions

Coexistence of different stableequilibrium points

Coexistence of stable equilibrium pointwith periodic attractor

Coexistence of 1-periodic cycle with2-periodic cycle

Coexistence of periodic attractor withchaotic attractor

stable persistently they are likely to be divergent oncebeyond the boundary of basins of attraction

4 Change in Robustness ofSupercavitating Vehicle

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

minus3

minus2

minus1

0

1

2

3120579 0

(rad)

0 5minus5z0 (m)

(a)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(b)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(c)

minus3

minus2

minus1

0

1

2

3

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

minus15

minus05

0

05

15120579 0

(rad)

minus1 0 1 2minus2z0 (m)

(a)

minus1 0 1 2minus2z0 (m)

minus15

minus05

0

05

15

120579 0(rad)

(b)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(c)

minus15

minus05

0

05

15

120579 0(rad)

minus1 0 1 2minus2z0 (m)

(d)

avoided to choose such variable combinations of parametersso as to improve robustness of the system

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

minus2

minus1

0

1

2120579 0

(rad)

0 5minus5z0 (m)

(a)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(b)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

0 5minus5z0 (m)

(d)

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

minus10

minus05

0

05

10120579 0

(rad)

minus05 0 05 10minus10z0 (m)

(a)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(b)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(c)

minus10

minus05

0

05

10

120579 0(rad)

minus05 0 05 10minus10z0 (m)

(d)

5 Conclusion

(1) In the system of supercavitating vehicle select anytwo parameters as variable parameters through

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Advances in Mathematical Physics

minus15

minus05

0

05

15120579 0

(rad)

minus05 0 05 15minus15z0 (m)

(a)

minus15

minus05

0

05

15

120579 0(rad)

minus05 0 05 15minus15z0 (m)

(b)

minus15

minus05

0

05

15

120579 0(rad)

0 5minus5z0 (m)

(c)

minus2

minus1

0

1

2

120579 0(rad)

minus2 0 2 4minus4z0 (m)

(d)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

the Fundamental Research Funds for the Central Universities(Grant no 30910612203)

References

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 13

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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