Research ArticleChaotic Motions of the Duffing-Van der Pol Oscillator withExternal and Parametric Excitations
Liangqiang Zhou and Fangqi Chen
Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China
Correspondence should be addressed to Liangqiang Zhou zlqrexsinacom and Fangqi Chen cfqyyfeyoucom
Received 14 May 2014 Revised 7 July 2014 Accepted 7 July 2014 Published 15 July 2014
Academic Editor Mickael Lallart
Copyright copy 2014 L Zhou and F ChenThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations are investigated both analyticallyand numerically in this paperThe critical curves separating the chaotic and nonchaotic regions are obtainedThe chaotic feature onthe system parameters is discussed in detail Some new dynamical phenomena including the controllable frequency are presentedfor this system Numerical results are given which verify the analytical ones
1 Introduction
Dufffing-Van der Pol oscillator has a wide usage in manyfields The dynamics of Duffing-Van der Pol oscillator hasbeen investigated widely in these years Using the Melnikovmethod Ravisankar et al [1] studied horseshoe chaos inDuffing-Van der Pol oscillator driven by different periodicforces Melnikov threshold curve was drawn in a parameterspace With the second-order averaging method and Mel-nikov method Jing et al [2] investigated chaotic motions inDuffing-Van der Pol equation with fifth nonlinear-restoringforce and two external forcing terms Numerical simulationswere given to show the consistence with the theoreticalanalysis and exhibit themore new complex dynamical behav-iors With the singularity analysis bifurcation propertiesof Duffing-Van der Pol system with two parameters undermultifrequency excitationswere studied byQin andChen [3]Using the residue harmonic method Leung et al [4] investi-gated periodic bifurcation of Duffing-Van der Pol oscillatorshaving fractional derivatives and time delay It was shownthat jumps and hysteresis phenomena can be delayed orremoved By using a simple transformation the first integralsand the solutions of the Duffing-Van der Pol type equationunder certain conditions were obtained by Udwadia and Cho[5] Using the residue harmonic homotopy a generalizedDuffing-Van der Pol oscillator with nonlinear fractional orderdamping was investigated by Leung et al [6] Nonlinear
dynamic behaviors of the harmonically forced oscillator werefurther explored by the harmonic balancemethod along withthe polynomial homotopy continuation technique By usingthe GYC partial region stability theory Ge and Li [7] studiedthe synchronization of new Mathieu-Van der Pol systemswith new Duffing-Van der Pol systems With the second-order averaging method the conditions for the existence andthe bifurcations of harmonics for the damped and drivenDuffing-Van der Pol systemwere obtained by Zhang et al [8]By the averaging method together with truncation of Taylorexpansions Li et al [9] investigated the dynamics of Duffing-Van der Pol oscillators under linear-plus-nonlinear positionfeedback control with two time delays By applying phasediagrams potential diagram Poincare maps bifurcationdiagrams and maximal Lyapunov exponent diagrams thenonlinear behavior and the complex state of the Duffing-Vander Pol equation with fifth nonlinear-restoring force and twoexternal periodic excitations were investigated by Shi et al[10] By using Melnikov analysis and numerical simulationsthe dynamical behaviors including chaos period-doublingcascades and strange attractors of the extended Duffing-Van der Pol system were investigated by Yu et al [11 12]With numerical methods Yu et al [13] also investigated thedynamical behavior of the extended Duffing-Van der Poloscillator with 1206016 potential Different routes to chaos and richdynamical phenomena were observed Patel and Sharma [14]revisited the stochastic Duffing-Van der Pol ldquofilteringrdquo in the
Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 131637 5 pageshttpdxdoiorg1011552014131637
2 Shock and Vibration
Fokker-Planck setting in lieu of the filtering in the Kushnersetting With the help of the modified quasiconservativeaveraging Li et al [15] investigated the stochastic responsesof Duffing-Van der Pol vibroimpact system under additivecolored noise excitation
In this paper the chaotic motions of the Duffing-Van derPol oscillator with external and parametric excitations arestudied analytically with the Melnikov method The criticalcurves separating the chaotic and nonchaotic regions areobtained The chaotic feature on the system parameters isdiscussed in detail and some new dynamical phenomena arepresentedNumerical simulations verify the analytical results
2 Formulation of the Problem
Consider the Duffing-Van der pol oscillator with external andparametric excitations
+ 119901 (1 minus 1199092) minus 120572119909 + 120573119909
3= 119865 (119905) (1)
where 119901 is a damping parameter 120572 gt 0 120573 gt 0 are realparameters and 119865(119905) = 119891(1 + 120575119909) cos120596119905 is the external andparametric forces
Assume the damping and excitation terms 119901 119891 are smallsetting 119901 = 120576119901 119891 = 120576119891 where 120576 is a small parameter then (1)can be written as
= 119910
119910 = 120572119909 minus 1205731199093+ 120576 [minus119901119910 (1 minus 119909
2) + 119891 (1 + 120575119909) cos120596119905]
(2)
Using the transformations
119909 = 119906radic120572
120573 119905 = radic
1
120572120591 (3)
then (2) can be written as
1199061015840= V
V1015840 = 119906 minus 1199063 minus 120576119901V(1 minus120572
where 119901 = 119901radic120572 119891 = 119891(120572radic120572radic120573) = 120596radic120572 and 1015840represents 119889119889120591
When 120576 = 0 the unperturbed system of (4) is
1199061015840= V
V1015840 = 119906 minus 1199063(5)
which is a planar Hamiltonian system with the Hamiltonian
119867(119906 V) =V2
2minus1199062
2+1199064
4 (6)
System (6) has three equilibrium points where (0 0) is asaddle point and (plusmn1 0) are all centers
minus1 10
u
Figure 1 The phase portrait of system (5)
There exist homoclinic orbits connecting (0 0) to itselfwith the expressions [16]
119906hom (120591) = plusmnradic2 sech (120591)
Vhom (120591) = ∓radic2 sech (120591) tanh (120591)(7)
and closed periodic orbits around (plusmn1 0)with the expressions[16]
119906119896 (120591) =
119896
radic21198962 minus 1119888119899(radic
1
21198962 minus 1120591 119896)
V119896 (120591) =
minus119896
radic21198962 minus 1119904119899(radic
1
21198962 minus 1120591 119896) 119889119899(radic
1
21198962 minus 1120591 119896)
(8)
see Figure 1 where 119904119899 119888119899 119889119899 are Jacobi elliptic functions and1radic2 lt 119896 lt 1 is the modulus of the Jacobi elliptic functionsThe period of the closed orbit is 119879
119896= 4radic21198962 minus 1119870(119896) where
119870(119896) is the complete elliptic integral of the first kind
3 Chaotic Motions of the System
Melnikov method [17] is an analytical tool to study chaoticsystems Recently chaotic motions of many systems forexample 1206016-Rayleigh oscillator [18] Duffing oscillator [19]Gyldenrsquos problem [20] and nonsmooth systems [21] havebeen investigated by theMelnikovmethod In this section weuse the Melnikov method to investigate the chaotic motionsof system (4) We compute the Melnikov functions of system(4) along the homoclinic orbit (7) as follows
119872(1205910) = int+infin
minusinfin
minus119901V2hom (120591) (1 minus120572
1205731199062
hom) (120591) 119889120591
+ int+infin
minusinfin
119891 cos (120591 + 1205910) Vhom (120591) 119889120591
Shock and Vibration 3
120572 = 112
10
8
6
4
2
0
0 1 2 3 4 5
120573 = 09
120573 = 07
120573 = 11
120573 = 13
120573 = 05
120573 = 03
120596
pf
(a)
120573 = 1
25
20
15
10
5
00 1 2 3 4 5
120572 = 13
120572 = 11
120572 = 09
120572 = 07
120572 = 05
120572 = 03
120596
pf
(b)
Figure 2 The critical curves for chaotic motions of system (2) in the case of 120575 = 0
+ 120575int+infin
minusinfin
119891radic120572
120573119906hom (120591) Vhom (120591) cos (120591 + 1205910) 119889120591
equiv minus1205831198680+ 119892 (119868
1+ 1205751198682) sin 120591
0
(9)
where
1198680= int+infin
minusinfin
V2hom (120591) (1 minus120572
1205731199062
hom (120591)) 119889120591 =4
3minus16120572
15120573
1198681= int+infin
minusinfin
Vhom (120591) sin 120591 119889120591 = radic2120587 sech(120587
2)
= radic2120587120596
radic120572sech( 120587120596
2radic120572)
1198682= int+infin
minusinfin
radic120572
120573119906hom (120591) Vhom (120591) sin 120591 119889120591
= minus2radic120572
120573int+infin
minusinfin
sech2 (120591) tanh2 (120591) sin 120591 119889120591
=2radic(120572120573)119890minus1205871205962radic120572 (1205962 minus 120572) 120587
(119890minus120587120596radic120572 minus 1) 120572
(10)
By Melnikov analysis we estimate the condition for trans-verse intersection and chaotic separatrix motion as follows
First taking 120575 = 0 which is the case of periodic externalexcitation letting 120572 = 1 for different values of 120573 we getthe critical curves separating the chaotic regions (below) andnonchaotic regions (above) as in Figure 2(a) Next Letting120573 = 1 the critical curves for different values of 120572 are shownin Figure 2(b)
Secondly taking 120575 = 1 which is the case of bothperiodic external and parametric excitations letting 120572 = 1for different values of 120573 we get the critical curves as inFigure 3(a) Next Letting 120573 = 1 the critical curves fordifferent values of 120572 are shown in Figure 3(b)
From Figures 2-3 we can obtain the following conclu-sions
(1) For the case of periodic external excitation the criticalcurves have the classical bell shape this means thatwith the excitations possessing sufficiently small orvery large periods the systems are not chaoticallyexcited When 120572 is fixed for each 120573 isin (0 08) thelarger the values of 120573 the larger the critical valuesfor chaotic motions while for 120573 gt 08 the larger thevalues of 120573 the smaller the critical values for chaoticmotions On the other hand when 120573 is fixed if 120572 lt09 for the case of small values of120596 that is the periodof the excitation is large the critical value for chaoticmotions decreases as 120572 increases when 120596 crosses acritical value the case is opposite so for the case oflarge values of120596 that is the period of the excitation issmall the critical value for chaotic motions increases
4 Shock and Vibration
120572 = 1
10
8
6
4
2
0
0 1 2 3 4 5
120573 = 13
120573 = 11
120573 = 09
120573 = 07
120573 = 05
120573 = 03
120596
pf
(a)
120573 = 1
0 1 2 3 4 5
120572 = 13
120572 = 11
120572 = 09
120572 = 07
120572 = 05
120572 = 03
120596
10
8
6
4
2
0
pf
(b)
Figure 3 The critical curves for chaotic motions of system (2) in the case of 120575 = 1
16
14
12
1
08
06
04
02
00 1 2 3
120596
pf
(a)
0 1 2 3
120596
10
8
6
4
2
0
pf
(b)
Figure 4 The theoretical and numerical critical values for chaos of system (2) in the case of (a) 120575 = 0 and (b) 120575 = 1
as120572 increases if120572 ge 09 for each excitation frequency120596 the critical value increases as 120572 increases
(2) For the case of parametric excitations the criti-cal curve first decreases quickly to zero and thenincreases at last it decreases to zero as 120596 increasesfrom zero There exists a controllable frequency 120596excited at which chaotic motions do not take place nomatter how large the excitation amplitude is When 120572
(120573) is fixed the controllable frequency increases as 120573(120572) increases
4 Numerical Simulations
First choosing the system parameters 120576 = 001 120572 = 1 120575 = 0120573 = 05 and 120596 isin (0 3) which is the case of external excita-tions the critical values119901119891 for chaoticmotions are shown in
Shock and Vibration 5
Figure 4(a) where the red curve is the theoretical predictionsand the black points are the numerical predictions
Next choosing the system parameters 120576 = 001 120572 = 1120575 = 1 120573 = 05 and 120596 isin (0 3) which is the case of bothexternal and parametric excitations the critical values 119901119891for chaotic motions are shown in Figure 4(b) where the redcurve is the theoretical predictions and the black points arethe numerical predictions From Figure 4 we can see that thedifference of the critical values for chaotic motions betweenthe theoretical and numerical predictions is very small sonumerical simulations agree with the analytical results
5 Conclusions
Using the Melnikov and numerical methods the chaoticmotions for the Duffing-Van der Pol oscillator with externaland parametrical excitations are investigated in this paperThe critical curves separating the chaotic and nonchaoticregions are obtained It is shown that there exists a control-lable frequency 120596 for the system with parametric excitationsWhen the system parameter 120572 (120573) is fixed the controllablefrequency increases as 120573 (120572) increases Some new dynamicalbehaviors are presented
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Theproject was supported byNational Science Foundation ofChina (11202095 and 11172125) China Postdoctoral ScienceFoundation (2013T60531) and the Science Foundation ofNUAA (NZ2013213)
References
[1] L Ravisankar V Ravichandran and V Chinnathambi ldquoPre-diction of horseshoe chaos in Duffing-Van der Pol oscillatordriven by different periodic forcesrdquo International Journal ofEngineering and Science vol 1 no 5 pp 17ndash25 2012
[2] Z Jing Z Yang and T Jiang ldquoComplex dynamics in Duffing-van der Pol equationrdquo Chaos Solitons and Fractals vol 27 no3 pp 722ndash747 2006
[3] Z Qin and Y Chen ldquoSingularity analysis of Duffing-vander Pol system with two bifurcation parameters under multi-frequency excitationsrdquo Applied Mathematics and MechanicsEnglish Edition vol 31 no 8 pp 1019ndash1026 2010
[4] A T Y Leung H X Yang and P Zhu ldquoPeriodic bifurcationof Duffing-van der Pol oscillators having fractional derivativesand time delayrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 40 pp 1013ndash1018 1993
[5] E F Udwadia and E Cho ldquoFirst integrals and solutionsof Duffing-Van der Pol type equationsrdquo Journal of AppliedMechanics-Transactions of the ASME vol 81 no 3 Article ID034501 2014
[6] A T Y Leung H X Yang and P Zhu ldquoNeimark bifurcationsof a generalized Duffing-Van der Pol oscillator with nonlinear
fractional order dampingrdquo International Journal of Bifurcationand Chaos vol 23 no 11 Article ID 1350177 19 pages 2013
[7] Z M Ge and S Y Li ldquoChaos generalized synchronization ofnew Mathieu-VAN der POL systems with new Duffing-VANder POL systems as functional system by GYC partial regionstability theoryrdquoAppliedMathematical Modelling vol 35 no 11pp 5245ndash5264 2011
[8] S G Zhang Z Y Zhu and Z Guo ldquoPrimary resonance andbifurcations in damped and drivenDuffing-Van der Pol systemrdquoAdvancedMaterials Research vol 216 no 1-2 pp 782ndash786 2011
[9] X Li H Zhang and L Zhang ldquoResponse of the Duffing-vander Pol oscillator under position feedback control with two timedelaysrdquo Shock and Vibration vol 18 no 1-2 pp 377ndash386 2011
[10] Y X Shi D Y Bai and W J Tao ldquoChaos and control ofthe Duffing-van der Pol equation with two external periodicexcitationsrdquo Journal of Hebei Normal University vol 34 no 6pp 631ndash635 2010
[11] J Yu W Pan and R Zhang ldquoPeriod-doubling cascades andstrange attractors in extended Duffing-Van der Pol oscillatorrdquoCommunications in Theoretical Physics vol 51 no 5 pp 865ndash868 2009
[12] J Yu and J R Li ldquoInvestigation on dynamics of the extendedDuffing-Van der Pol systemrdquo Zeitschrift fur Naturforschung A AJournal of Physical Sciences vol 64 no 5-6 pp 341ndash346 2009
[13] J Yu Z Xie and L Yu ldquoComplex dynamics in aDuffing-Van derPol oscillator with 1205956 potentialrdquo Journal of the Physical Societyof Japan vol 77 no 11 Article ID 114003 2008
[14] H G Patel and S N Sharma ldquoFiltering for a Duffing-van derPol stochastic differential equationrdquo Applied Mathematics andComputation vol 226 pp 386ndash397 2014
[15] C Li W Xu L Wang and D X Li ldquoStochastic responsesof Duffing-Van der Pol vibro-impact system under additivecolored noise excitationrdquo Chinese Physics B vol 22 no 11Article ID 110205 2013
[16] J B Li Chaos and Melnikov Method Chongqing UniversityPress Chongqing China 1989 (Chinese)
[17] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems andChaos vol 2 ofTexts in AppliedMathematics SpringerNew York NY USA 1990
[18] M S Siewe C Tchawoua and S Rajasekar ldquoHomoclinicbifurcation and chaos in 1206016-Rayleigh oscillator with three wellsdriven by an amplitude modulated forcerdquo International Journalof Bifurcation and Chaos vol 21 no 6 pp 1583ndash1593 2011
[19] V Ravichandran V Chinnathambi and S Rajasekar ldquoHomo-clinic bifurcation and chaos in Duffing oscillator driven by anamplitude-modulated forcerdquo Physica A Statistical Mechanicsand its Applications vol 376 pp 223ndash236 2007
[20] C Castilho and M Marchesin ldquoA practical use of the Melnikovhomoclinic methodrdquo Journal of Mathematical Physics vol 50no 11 Article ID 112704 11 pages 2009
[21] L Shi Y Zou and T Kupper ldquoMelnikov method and detectionof chaos for non-smooth systemsrdquo Acta Mathematicae Appli-catae Sinica English Series vol 29 no 4 pp 881ndash896 2013
Fokker-Planck setting in lieu of the filtering in the Kushnersetting With the help of the modified quasiconservativeaveraging Li et al [15] investigated the stochastic responsesof Duffing-Van der Pol vibroimpact system under additivecolored noise excitation
In this paper the chaotic motions of the Duffing-Van derPol oscillator with external and parametric excitations arestudied analytically with the Melnikov method The criticalcurves separating the chaotic and nonchaotic regions areobtained The chaotic feature on the system parameters isdiscussed in detail and some new dynamical phenomena arepresentedNumerical simulations verify the analytical results
2 Formulation of the Problem
Consider the Duffing-Van der pol oscillator with external andparametric excitations
+ 119901 (1 minus 1199092) minus 120572119909 + 120573119909
3= 119865 (119905) (1)
where 119901 is a damping parameter 120572 gt 0 120573 gt 0 are realparameters and 119865(119905) = 119891(1 + 120575119909) cos120596119905 is the external andparametric forces
Assume the damping and excitation terms 119901 119891 are smallsetting 119901 = 120576119901 119891 = 120576119891 where 120576 is a small parameter then (1)can be written as
= 119910
119910 = 120572119909 minus 1205731199093+ 120576 [minus119901119910 (1 minus 119909
2) + 119891 (1 + 120575119909) cos120596119905]
(2)
Using the transformations
119909 = 119906radic120572
120573 119905 = radic
1
120572120591 (3)
then (2) can be written as
1199061015840= V
V1015840 = 119906 minus 1199063 minus 120576119901V(1 minus120572
where 119901 = 119901radic120572 119891 = 119891(120572radic120572radic120573) = 120596radic120572 and 1015840represents 119889119889120591
When 120576 = 0 the unperturbed system of (4) is
1199061015840= V
V1015840 = 119906 minus 1199063(5)
which is a planar Hamiltonian system with the Hamiltonian
119867(119906 V) =V2
2minus1199062
2+1199064
4 (6)
System (6) has three equilibrium points where (0 0) is asaddle point and (plusmn1 0) are all centers
minus1 10
u
Figure 1 The phase portrait of system (5)
There exist homoclinic orbits connecting (0 0) to itselfwith the expressions [16]
119906hom (120591) = plusmnradic2 sech (120591)
Vhom (120591) = ∓radic2 sech (120591) tanh (120591)(7)
and closed periodic orbits around (plusmn1 0)with the expressions[16]
119906119896 (120591) =
119896
radic21198962 minus 1119888119899(radic
1
21198962 minus 1120591 119896)
V119896 (120591) =
minus119896
radic21198962 minus 1119904119899(radic
1
21198962 minus 1120591 119896) 119889119899(radic
1
21198962 minus 1120591 119896)
(8)
see Figure 1 where 119904119899 119888119899 119889119899 are Jacobi elliptic functions and1radic2 lt 119896 lt 1 is the modulus of the Jacobi elliptic functionsThe period of the closed orbit is 119879
119896= 4radic21198962 minus 1119870(119896) where
119870(119896) is the complete elliptic integral of the first kind
3 Chaotic Motions of the System
Melnikov method [17] is an analytical tool to study chaoticsystems Recently chaotic motions of many systems forexample 1206016-Rayleigh oscillator [18] Duffing oscillator [19]Gyldenrsquos problem [20] and nonsmooth systems [21] havebeen investigated by theMelnikovmethod In this section weuse the Melnikov method to investigate the chaotic motionsof system (4) We compute the Melnikov functions of system(4) along the homoclinic orbit (7) as follows
119872(1205910) = int+infin
minusinfin
minus119901V2hom (120591) (1 minus120572
1205731199062
hom) (120591) 119889120591
+ int+infin
minusinfin
119891 cos (120591 + 1205910) Vhom (120591) 119889120591
Shock and Vibration 3
120572 = 112
10
8
6
4
2
0
0 1 2 3 4 5
120573 = 09
120573 = 07
120573 = 11
120573 = 13
120573 = 05
120573 = 03
120596
pf
(a)
120573 = 1
25
20
15
10
5
00 1 2 3 4 5
120572 = 13
120572 = 11
120572 = 09
120572 = 07
120572 = 05
120572 = 03
120596
pf
(b)
Figure 2 The critical curves for chaotic motions of system (2) in the case of 120575 = 0
+ 120575int+infin
minusinfin
119891radic120572
120573119906hom (120591) Vhom (120591) cos (120591 + 1205910) 119889120591
equiv minus1205831198680+ 119892 (119868
1+ 1205751198682) sin 120591
0
(9)
where
1198680= int+infin
minusinfin
V2hom (120591) (1 minus120572
1205731199062
hom (120591)) 119889120591 =4
3minus16120572
15120573
1198681= int+infin
minusinfin
Vhom (120591) sin 120591 119889120591 = radic2120587 sech(120587
2)
= radic2120587120596
radic120572sech( 120587120596
2radic120572)
1198682= int+infin
minusinfin
radic120572
120573119906hom (120591) Vhom (120591) sin 120591 119889120591
= minus2radic120572
120573int+infin
minusinfin
sech2 (120591) tanh2 (120591) sin 120591 119889120591
=2radic(120572120573)119890minus1205871205962radic120572 (1205962 minus 120572) 120587
(119890minus120587120596radic120572 minus 1) 120572
(10)
By Melnikov analysis we estimate the condition for trans-verse intersection and chaotic separatrix motion as follows
First taking 120575 = 0 which is the case of periodic externalexcitation letting 120572 = 1 for different values of 120573 we getthe critical curves separating the chaotic regions (below) andnonchaotic regions (above) as in Figure 2(a) Next Letting120573 = 1 the critical curves for different values of 120572 are shownin Figure 2(b)
Secondly taking 120575 = 1 which is the case of bothperiodic external and parametric excitations letting 120572 = 1for different values of 120573 we get the critical curves as inFigure 3(a) Next Letting 120573 = 1 the critical curves fordifferent values of 120572 are shown in Figure 3(b)
From Figures 2-3 we can obtain the following conclu-sions
(1) For the case of periodic external excitation the criticalcurves have the classical bell shape this means thatwith the excitations possessing sufficiently small orvery large periods the systems are not chaoticallyexcited When 120572 is fixed for each 120573 isin (0 08) thelarger the values of 120573 the larger the critical valuesfor chaotic motions while for 120573 gt 08 the larger thevalues of 120573 the smaller the critical values for chaoticmotions On the other hand when 120573 is fixed if 120572 lt09 for the case of small values of120596 that is the periodof the excitation is large the critical value for chaoticmotions decreases as 120572 increases when 120596 crosses acritical value the case is opposite so for the case oflarge values of120596 that is the period of the excitation issmall the critical value for chaotic motions increases
4 Shock and Vibration
120572 = 1
10
8
6
4
2
0
0 1 2 3 4 5
120573 = 13
120573 = 11
120573 = 09
120573 = 07
120573 = 05
120573 = 03
120596
pf
(a)
120573 = 1
0 1 2 3 4 5
120572 = 13
120572 = 11
120572 = 09
120572 = 07
120572 = 05
120572 = 03
120596
10
8
6
4
2
0
pf
(b)
Figure 3 The critical curves for chaotic motions of system (2) in the case of 120575 = 1
16
14
12
1
08
06
04
02
00 1 2 3
120596
pf
(a)
0 1 2 3
120596
10
8
6
4
2
0
pf
(b)
Figure 4 The theoretical and numerical critical values for chaos of system (2) in the case of (a) 120575 = 0 and (b) 120575 = 1
as120572 increases if120572 ge 09 for each excitation frequency120596 the critical value increases as 120572 increases
(2) For the case of parametric excitations the criti-cal curve first decreases quickly to zero and thenincreases at last it decreases to zero as 120596 increasesfrom zero There exists a controllable frequency 120596excited at which chaotic motions do not take place nomatter how large the excitation amplitude is When 120572
(120573) is fixed the controllable frequency increases as 120573(120572) increases
4 Numerical Simulations
First choosing the system parameters 120576 = 001 120572 = 1 120575 = 0120573 = 05 and 120596 isin (0 3) which is the case of external excita-tions the critical values119901119891 for chaoticmotions are shown in
Shock and Vibration 5
Figure 4(a) where the red curve is the theoretical predictionsand the black points are the numerical predictions
Next choosing the system parameters 120576 = 001 120572 = 1120575 = 1 120573 = 05 and 120596 isin (0 3) which is the case of bothexternal and parametric excitations the critical values 119901119891for chaotic motions are shown in Figure 4(b) where the redcurve is the theoretical predictions and the black points arethe numerical predictions From Figure 4 we can see that thedifference of the critical values for chaotic motions betweenthe theoretical and numerical predictions is very small sonumerical simulations agree with the analytical results
5 Conclusions
Using the Melnikov and numerical methods the chaoticmotions for the Duffing-Van der Pol oscillator with externaland parametrical excitations are investigated in this paperThe critical curves separating the chaotic and nonchaoticregions are obtained It is shown that there exists a control-lable frequency 120596 for the system with parametric excitationsWhen the system parameter 120572 (120573) is fixed the controllablefrequency increases as 120573 (120572) increases Some new dynamicalbehaviors are presented
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Theproject was supported byNational Science Foundation ofChina (11202095 and 11172125) China Postdoctoral ScienceFoundation (2013T60531) and the Science Foundation ofNUAA (NZ2013213)
References
[1] L Ravisankar V Ravichandran and V Chinnathambi ldquoPre-diction of horseshoe chaos in Duffing-Van der Pol oscillatordriven by different periodic forcesrdquo International Journal ofEngineering and Science vol 1 no 5 pp 17ndash25 2012
[2] Z Jing Z Yang and T Jiang ldquoComplex dynamics in Duffing-van der Pol equationrdquo Chaos Solitons and Fractals vol 27 no3 pp 722ndash747 2006
[3] Z Qin and Y Chen ldquoSingularity analysis of Duffing-vander Pol system with two bifurcation parameters under multi-frequency excitationsrdquo Applied Mathematics and MechanicsEnglish Edition vol 31 no 8 pp 1019ndash1026 2010
[4] A T Y Leung H X Yang and P Zhu ldquoPeriodic bifurcationof Duffing-van der Pol oscillators having fractional derivativesand time delayrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 40 pp 1013ndash1018 1993
[5] E F Udwadia and E Cho ldquoFirst integrals and solutionsof Duffing-Van der Pol type equationsrdquo Journal of AppliedMechanics-Transactions of the ASME vol 81 no 3 Article ID034501 2014
[6] A T Y Leung H X Yang and P Zhu ldquoNeimark bifurcationsof a generalized Duffing-Van der Pol oscillator with nonlinear
fractional order dampingrdquo International Journal of Bifurcationand Chaos vol 23 no 11 Article ID 1350177 19 pages 2013
[7] Z M Ge and S Y Li ldquoChaos generalized synchronization ofnew Mathieu-VAN der POL systems with new Duffing-VANder POL systems as functional system by GYC partial regionstability theoryrdquoAppliedMathematical Modelling vol 35 no 11pp 5245ndash5264 2011
[8] S G Zhang Z Y Zhu and Z Guo ldquoPrimary resonance andbifurcations in damped and drivenDuffing-Van der Pol systemrdquoAdvancedMaterials Research vol 216 no 1-2 pp 782ndash786 2011
[9] X Li H Zhang and L Zhang ldquoResponse of the Duffing-vander Pol oscillator under position feedback control with two timedelaysrdquo Shock and Vibration vol 18 no 1-2 pp 377ndash386 2011
[10] Y X Shi D Y Bai and W J Tao ldquoChaos and control ofthe Duffing-van der Pol equation with two external periodicexcitationsrdquo Journal of Hebei Normal University vol 34 no 6pp 631ndash635 2010
[11] J Yu W Pan and R Zhang ldquoPeriod-doubling cascades andstrange attractors in extended Duffing-Van der Pol oscillatorrdquoCommunications in Theoretical Physics vol 51 no 5 pp 865ndash868 2009
[12] J Yu and J R Li ldquoInvestigation on dynamics of the extendedDuffing-Van der Pol systemrdquo Zeitschrift fur Naturforschung A AJournal of Physical Sciences vol 64 no 5-6 pp 341ndash346 2009
[13] J Yu Z Xie and L Yu ldquoComplex dynamics in aDuffing-Van derPol oscillator with 1205956 potentialrdquo Journal of the Physical Societyof Japan vol 77 no 11 Article ID 114003 2008
[14] H G Patel and S N Sharma ldquoFiltering for a Duffing-van derPol stochastic differential equationrdquo Applied Mathematics andComputation vol 226 pp 386ndash397 2014
[15] C Li W Xu L Wang and D X Li ldquoStochastic responsesof Duffing-Van der Pol vibro-impact system under additivecolored noise excitationrdquo Chinese Physics B vol 22 no 11Article ID 110205 2013
[16] J B Li Chaos and Melnikov Method Chongqing UniversityPress Chongqing China 1989 (Chinese)
[17] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems andChaos vol 2 ofTexts in AppliedMathematics SpringerNew York NY USA 1990
[18] M S Siewe C Tchawoua and S Rajasekar ldquoHomoclinicbifurcation and chaos in 1206016-Rayleigh oscillator with three wellsdriven by an amplitude modulated forcerdquo International Journalof Bifurcation and Chaos vol 21 no 6 pp 1583ndash1593 2011
[19] V Ravichandran V Chinnathambi and S Rajasekar ldquoHomo-clinic bifurcation and chaos in Duffing oscillator driven by anamplitude-modulated forcerdquo Physica A Statistical Mechanicsand its Applications vol 376 pp 223ndash236 2007
[20] C Castilho and M Marchesin ldquoA practical use of the Melnikovhomoclinic methodrdquo Journal of Mathematical Physics vol 50no 11 Article ID 112704 11 pages 2009
[21] L Shi Y Zou and T Kupper ldquoMelnikov method and detectionof chaos for non-smooth systemsrdquo Acta Mathematicae Appli-catae Sinica English Series vol 29 no 4 pp 881ndash896 2013
First taking 120575 = 0 which is the case of periodic externalexcitation letting 120572 = 1 for different values of 120573 we getthe critical curves separating the chaotic regions (below) andnonchaotic regions (above) as in Figure 2(a) Next Letting120573 = 1 the critical curves for different values of 120572 are shownin Figure 2(b)
Secondly taking 120575 = 1 which is the case of bothperiodic external and parametric excitations letting 120572 = 1for different values of 120573 we get the critical curves as inFigure 3(a) Next Letting 120573 = 1 the critical curves fordifferent values of 120572 are shown in Figure 3(b)
From Figures 2-3 we can obtain the following conclu-sions
(1) For the case of periodic external excitation the criticalcurves have the classical bell shape this means thatwith the excitations possessing sufficiently small orvery large periods the systems are not chaoticallyexcited When 120572 is fixed for each 120573 isin (0 08) thelarger the values of 120573 the larger the critical valuesfor chaotic motions while for 120573 gt 08 the larger thevalues of 120573 the smaller the critical values for chaoticmotions On the other hand when 120573 is fixed if 120572 lt09 for the case of small values of120596 that is the periodof the excitation is large the critical value for chaoticmotions decreases as 120572 increases when 120596 crosses acritical value the case is opposite so for the case oflarge values of120596 that is the period of the excitation issmall the critical value for chaotic motions increases
4 Shock and Vibration
120572 = 1
10
8
6
4
2
0
0 1 2 3 4 5
120573 = 13
120573 = 11
120573 = 09
120573 = 07
120573 = 05
120573 = 03
120596
pf
(a)
120573 = 1
0 1 2 3 4 5
120572 = 13
120572 = 11
120572 = 09
120572 = 07
120572 = 05
120572 = 03
120596
10
8
6
4
2
0
pf
(b)
Figure 3 The critical curves for chaotic motions of system (2) in the case of 120575 = 1
16
14
12
1
08
06
04
02
00 1 2 3
120596
pf
(a)
0 1 2 3
120596
10
8
6
4
2
0
pf
(b)
Figure 4 The theoretical and numerical critical values for chaos of system (2) in the case of (a) 120575 = 0 and (b) 120575 = 1
as120572 increases if120572 ge 09 for each excitation frequency120596 the critical value increases as 120572 increases
(2) For the case of parametric excitations the criti-cal curve first decreases quickly to zero and thenincreases at last it decreases to zero as 120596 increasesfrom zero There exists a controllable frequency 120596excited at which chaotic motions do not take place nomatter how large the excitation amplitude is When 120572
(120573) is fixed the controllable frequency increases as 120573(120572) increases
4 Numerical Simulations
First choosing the system parameters 120576 = 001 120572 = 1 120575 = 0120573 = 05 and 120596 isin (0 3) which is the case of external excita-tions the critical values119901119891 for chaoticmotions are shown in
Shock and Vibration 5
Figure 4(a) where the red curve is the theoretical predictionsand the black points are the numerical predictions
Next choosing the system parameters 120576 = 001 120572 = 1120575 = 1 120573 = 05 and 120596 isin (0 3) which is the case of bothexternal and parametric excitations the critical values 119901119891for chaotic motions are shown in Figure 4(b) where the redcurve is the theoretical predictions and the black points arethe numerical predictions From Figure 4 we can see that thedifference of the critical values for chaotic motions betweenthe theoretical and numerical predictions is very small sonumerical simulations agree with the analytical results
5 Conclusions
Using the Melnikov and numerical methods the chaoticmotions for the Duffing-Van der Pol oscillator with externaland parametrical excitations are investigated in this paperThe critical curves separating the chaotic and nonchaoticregions are obtained It is shown that there exists a control-lable frequency 120596 for the system with parametric excitationsWhen the system parameter 120572 (120573) is fixed the controllablefrequency increases as 120573 (120572) increases Some new dynamicalbehaviors are presented
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Theproject was supported byNational Science Foundation ofChina (11202095 and 11172125) China Postdoctoral ScienceFoundation (2013T60531) and the Science Foundation ofNUAA (NZ2013213)
References
[1] L Ravisankar V Ravichandran and V Chinnathambi ldquoPre-diction of horseshoe chaos in Duffing-Van der Pol oscillatordriven by different periodic forcesrdquo International Journal ofEngineering and Science vol 1 no 5 pp 17ndash25 2012
[2] Z Jing Z Yang and T Jiang ldquoComplex dynamics in Duffing-van der Pol equationrdquo Chaos Solitons and Fractals vol 27 no3 pp 722ndash747 2006
[3] Z Qin and Y Chen ldquoSingularity analysis of Duffing-vander Pol system with two bifurcation parameters under multi-frequency excitationsrdquo Applied Mathematics and MechanicsEnglish Edition vol 31 no 8 pp 1019ndash1026 2010
[4] A T Y Leung H X Yang and P Zhu ldquoPeriodic bifurcationof Duffing-van der Pol oscillators having fractional derivativesand time delayrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 40 pp 1013ndash1018 1993
[5] E F Udwadia and E Cho ldquoFirst integrals and solutionsof Duffing-Van der Pol type equationsrdquo Journal of AppliedMechanics-Transactions of the ASME vol 81 no 3 Article ID034501 2014
[6] A T Y Leung H X Yang and P Zhu ldquoNeimark bifurcationsof a generalized Duffing-Van der Pol oscillator with nonlinear
fractional order dampingrdquo International Journal of Bifurcationand Chaos vol 23 no 11 Article ID 1350177 19 pages 2013
[7] Z M Ge and S Y Li ldquoChaos generalized synchronization ofnew Mathieu-VAN der POL systems with new Duffing-VANder POL systems as functional system by GYC partial regionstability theoryrdquoAppliedMathematical Modelling vol 35 no 11pp 5245ndash5264 2011
[8] S G Zhang Z Y Zhu and Z Guo ldquoPrimary resonance andbifurcations in damped and drivenDuffing-Van der Pol systemrdquoAdvancedMaterials Research vol 216 no 1-2 pp 782ndash786 2011
[9] X Li H Zhang and L Zhang ldquoResponse of the Duffing-vander Pol oscillator under position feedback control with two timedelaysrdquo Shock and Vibration vol 18 no 1-2 pp 377ndash386 2011
[10] Y X Shi D Y Bai and W J Tao ldquoChaos and control ofthe Duffing-van der Pol equation with two external periodicexcitationsrdquo Journal of Hebei Normal University vol 34 no 6pp 631ndash635 2010
[11] J Yu W Pan and R Zhang ldquoPeriod-doubling cascades andstrange attractors in extended Duffing-Van der Pol oscillatorrdquoCommunications in Theoretical Physics vol 51 no 5 pp 865ndash868 2009
[12] J Yu and J R Li ldquoInvestigation on dynamics of the extendedDuffing-Van der Pol systemrdquo Zeitschrift fur Naturforschung A AJournal of Physical Sciences vol 64 no 5-6 pp 341ndash346 2009
[13] J Yu Z Xie and L Yu ldquoComplex dynamics in aDuffing-Van derPol oscillator with 1205956 potentialrdquo Journal of the Physical Societyof Japan vol 77 no 11 Article ID 114003 2008
[14] H G Patel and S N Sharma ldquoFiltering for a Duffing-van derPol stochastic differential equationrdquo Applied Mathematics andComputation vol 226 pp 386ndash397 2014
[15] C Li W Xu L Wang and D X Li ldquoStochastic responsesof Duffing-Van der Pol vibro-impact system under additivecolored noise excitationrdquo Chinese Physics B vol 22 no 11Article ID 110205 2013
[16] J B Li Chaos and Melnikov Method Chongqing UniversityPress Chongqing China 1989 (Chinese)
[17] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems andChaos vol 2 ofTexts in AppliedMathematics SpringerNew York NY USA 1990
[18] M S Siewe C Tchawoua and S Rajasekar ldquoHomoclinicbifurcation and chaos in 1206016-Rayleigh oscillator with three wellsdriven by an amplitude modulated forcerdquo International Journalof Bifurcation and Chaos vol 21 no 6 pp 1583ndash1593 2011
[19] V Ravichandran V Chinnathambi and S Rajasekar ldquoHomo-clinic bifurcation and chaos in Duffing oscillator driven by anamplitude-modulated forcerdquo Physica A Statistical Mechanicsand its Applications vol 376 pp 223ndash236 2007
[20] C Castilho and M Marchesin ldquoA practical use of the Melnikovhomoclinic methodrdquo Journal of Mathematical Physics vol 50no 11 Article ID 112704 11 pages 2009
[21] L Shi Y Zou and T Kupper ldquoMelnikov method and detectionof chaos for non-smooth systemsrdquo Acta Mathematicae Appli-catae Sinica English Series vol 29 no 4 pp 881ndash896 2013
Figure 3 The critical curves for chaotic motions of system (2) in the case of 120575 = 1
16
14
12
1
08
06
04
02
00 1 2 3
120596
pf
(a)
0 1 2 3
120596
10
8
6
4
2
0
pf
(b)
Figure 4 The theoretical and numerical critical values for chaos of system (2) in the case of (a) 120575 = 0 and (b) 120575 = 1
as120572 increases if120572 ge 09 for each excitation frequency120596 the critical value increases as 120572 increases
(2) For the case of parametric excitations the criti-cal curve first decreases quickly to zero and thenincreases at last it decreases to zero as 120596 increasesfrom zero There exists a controllable frequency 120596excited at which chaotic motions do not take place nomatter how large the excitation amplitude is When 120572
(120573) is fixed the controllable frequency increases as 120573(120572) increases
4 Numerical Simulations
First choosing the system parameters 120576 = 001 120572 = 1 120575 = 0120573 = 05 and 120596 isin (0 3) which is the case of external excita-tions the critical values119901119891 for chaoticmotions are shown in
Shock and Vibration 5
Figure 4(a) where the red curve is the theoretical predictionsand the black points are the numerical predictions
Next choosing the system parameters 120576 = 001 120572 = 1120575 = 1 120573 = 05 and 120596 isin (0 3) which is the case of bothexternal and parametric excitations the critical values 119901119891for chaotic motions are shown in Figure 4(b) where the redcurve is the theoretical predictions and the black points arethe numerical predictions From Figure 4 we can see that thedifference of the critical values for chaotic motions betweenthe theoretical and numerical predictions is very small sonumerical simulations agree with the analytical results
5 Conclusions
Using the Melnikov and numerical methods the chaoticmotions for the Duffing-Van der Pol oscillator with externaland parametrical excitations are investigated in this paperThe critical curves separating the chaotic and nonchaoticregions are obtained It is shown that there exists a control-lable frequency 120596 for the system with parametric excitationsWhen the system parameter 120572 (120573) is fixed the controllablefrequency increases as 120573 (120572) increases Some new dynamicalbehaviors are presented
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Theproject was supported byNational Science Foundation ofChina (11202095 and 11172125) China Postdoctoral ScienceFoundation (2013T60531) and the Science Foundation ofNUAA (NZ2013213)
References
[1] L Ravisankar V Ravichandran and V Chinnathambi ldquoPre-diction of horseshoe chaos in Duffing-Van der Pol oscillatordriven by different periodic forcesrdquo International Journal ofEngineering and Science vol 1 no 5 pp 17ndash25 2012
[2] Z Jing Z Yang and T Jiang ldquoComplex dynamics in Duffing-van der Pol equationrdquo Chaos Solitons and Fractals vol 27 no3 pp 722ndash747 2006
[3] Z Qin and Y Chen ldquoSingularity analysis of Duffing-vander Pol system with two bifurcation parameters under multi-frequency excitationsrdquo Applied Mathematics and MechanicsEnglish Edition vol 31 no 8 pp 1019ndash1026 2010
[4] A T Y Leung H X Yang and P Zhu ldquoPeriodic bifurcationof Duffing-van der Pol oscillators having fractional derivativesand time delayrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 40 pp 1013ndash1018 1993
[5] E F Udwadia and E Cho ldquoFirst integrals and solutionsof Duffing-Van der Pol type equationsrdquo Journal of AppliedMechanics-Transactions of the ASME vol 81 no 3 Article ID034501 2014
[6] A T Y Leung H X Yang and P Zhu ldquoNeimark bifurcationsof a generalized Duffing-Van der Pol oscillator with nonlinear
fractional order dampingrdquo International Journal of Bifurcationand Chaos vol 23 no 11 Article ID 1350177 19 pages 2013
[7] Z M Ge and S Y Li ldquoChaos generalized synchronization ofnew Mathieu-VAN der POL systems with new Duffing-VANder POL systems as functional system by GYC partial regionstability theoryrdquoAppliedMathematical Modelling vol 35 no 11pp 5245ndash5264 2011
[8] S G Zhang Z Y Zhu and Z Guo ldquoPrimary resonance andbifurcations in damped and drivenDuffing-Van der Pol systemrdquoAdvancedMaterials Research vol 216 no 1-2 pp 782ndash786 2011
[9] X Li H Zhang and L Zhang ldquoResponse of the Duffing-vander Pol oscillator under position feedback control with two timedelaysrdquo Shock and Vibration vol 18 no 1-2 pp 377ndash386 2011
[10] Y X Shi D Y Bai and W J Tao ldquoChaos and control ofthe Duffing-van der Pol equation with two external periodicexcitationsrdquo Journal of Hebei Normal University vol 34 no 6pp 631ndash635 2010
[11] J Yu W Pan and R Zhang ldquoPeriod-doubling cascades andstrange attractors in extended Duffing-Van der Pol oscillatorrdquoCommunications in Theoretical Physics vol 51 no 5 pp 865ndash868 2009
[12] J Yu and J R Li ldquoInvestigation on dynamics of the extendedDuffing-Van der Pol systemrdquo Zeitschrift fur Naturforschung A AJournal of Physical Sciences vol 64 no 5-6 pp 341ndash346 2009
[13] J Yu Z Xie and L Yu ldquoComplex dynamics in aDuffing-Van derPol oscillator with 1205956 potentialrdquo Journal of the Physical Societyof Japan vol 77 no 11 Article ID 114003 2008
[14] H G Patel and S N Sharma ldquoFiltering for a Duffing-van derPol stochastic differential equationrdquo Applied Mathematics andComputation vol 226 pp 386ndash397 2014
[15] C Li W Xu L Wang and D X Li ldquoStochastic responsesof Duffing-Van der Pol vibro-impact system under additivecolored noise excitationrdquo Chinese Physics B vol 22 no 11Article ID 110205 2013
[16] J B Li Chaos and Melnikov Method Chongqing UniversityPress Chongqing China 1989 (Chinese)
[17] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems andChaos vol 2 ofTexts in AppliedMathematics SpringerNew York NY USA 1990
[18] M S Siewe C Tchawoua and S Rajasekar ldquoHomoclinicbifurcation and chaos in 1206016-Rayleigh oscillator with three wellsdriven by an amplitude modulated forcerdquo International Journalof Bifurcation and Chaos vol 21 no 6 pp 1583ndash1593 2011
[19] V Ravichandran V Chinnathambi and S Rajasekar ldquoHomo-clinic bifurcation and chaos in Duffing oscillator driven by anamplitude-modulated forcerdquo Physica A Statistical Mechanicsand its Applications vol 376 pp 223ndash236 2007
[20] C Castilho and M Marchesin ldquoA practical use of the Melnikovhomoclinic methodrdquo Journal of Mathematical Physics vol 50no 11 Article ID 112704 11 pages 2009
[21] L Shi Y Zou and T Kupper ldquoMelnikov method and detectionof chaos for non-smooth systemsrdquo Acta Mathematicae Appli-catae Sinica English Series vol 29 no 4 pp 881ndash896 2013
Figure 4(a) where the red curve is the theoretical predictionsand the black points are the numerical predictions
Next choosing the system parameters 120576 = 001 120572 = 1120575 = 1 120573 = 05 and 120596 isin (0 3) which is the case of bothexternal and parametric excitations the critical values 119901119891for chaotic motions are shown in Figure 4(b) where the redcurve is the theoretical predictions and the black points arethe numerical predictions From Figure 4 we can see that thedifference of the critical values for chaotic motions betweenthe theoretical and numerical predictions is very small sonumerical simulations agree with the analytical results
5 Conclusions
Using the Melnikov and numerical methods the chaoticmotions for the Duffing-Van der Pol oscillator with externaland parametrical excitations are investigated in this paperThe critical curves separating the chaotic and nonchaoticregions are obtained It is shown that there exists a control-lable frequency 120596 for the system with parametric excitationsWhen the system parameter 120572 (120573) is fixed the controllablefrequency increases as 120573 (120572) increases Some new dynamicalbehaviors are presented
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Theproject was supported byNational Science Foundation ofChina (11202095 and 11172125) China Postdoctoral ScienceFoundation (2013T60531) and the Science Foundation ofNUAA (NZ2013213)
References
[1] L Ravisankar V Ravichandran and V Chinnathambi ldquoPre-diction of horseshoe chaos in Duffing-Van der Pol oscillatordriven by different periodic forcesrdquo International Journal ofEngineering and Science vol 1 no 5 pp 17ndash25 2012
[2] Z Jing Z Yang and T Jiang ldquoComplex dynamics in Duffing-van der Pol equationrdquo Chaos Solitons and Fractals vol 27 no3 pp 722ndash747 2006
[3] Z Qin and Y Chen ldquoSingularity analysis of Duffing-vander Pol system with two bifurcation parameters under multi-frequency excitationsrdquo Applied Mathematics and MechanicsEnglish Edition vol 31 no 8 pp 1019ndash1026 2010
[4] A T Y Leung H X Yang and P Zhu ldquoPeriodic bifurcationof Duffing-van der Pol oscillators having fractional derivativesand time delayrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 40 pp 1013ndash1018 1993
[5] E F Udwadia and E Cho ldquoFirst integrals and solutionsof Duffing-Van der Pol type equationsrdquo Journal of AppliedMechanics-Transactions of the ASME vol 81 no 3 Article ID034501 2014
[6] A T Y Leung H X Yang and P Zhu ldquoNeimark bifurcationsof a generalized Duffing-Van der Pol oscillator with nonlinear
fractional order dampingrdquo International Journal of Bifurcationand Chaos vol 23 no 11 Article ID 1350177 19 pages 2013
[7] Z M Ge and S Y Li ldquoChaos generalized synchronization ofnew Mathieu-VAN der POL systems with new Duffing-VANder POL systems as functional system by GYC partial regionstability theoryrdquoAppliedMathematical Modelling vol 35 no 11pp 5245ndash5264 2011
[8] S G Zhang Z Y Zhu and Z Guo ldquoPrimary resonance andbifurcations in damped and drivenDuffing-Van der Pol systemrdquoAdvancedMaterials Research vol 216 no 1-2 pp 782ndash786 2011
[9] X Li H Zhang and L Zhang ldquoResponse of the Duffing-vander Pol oscillator under position feedback control with two timedelaysrdquo Shock and Vibration vol 18 no 1-2 pp 377ndash386 2011
[10] Y X Shi D Y Bai and W J Tao ldquoChaos and control ofthe Duffing-van der Pol equation with two external periodicexcitationsrdquo Journal of Hebei Normal University vol 34 no 6pp 631ndash635 2010
[11] J Yu W Pan and R Zhang ldquoPeriod-doubling cascades andstrange attractors in extended Duffing-Van der Pol oscillatorrdquoCommunications in Theoretical Physics vol 51 no 5 pp 865ndash868 2009
[12] J Yu and J R Li ldquoInvestigation on dynamics of the extendedDuffing-Van der Pol systemrdquo Zeitschrift fur Naturforschung A AJournal of Physical Sciences vol 64 no 5-6 pp 341ndash346 2009
[13] J Yu Z Xie and L Yu ldquoComplex dynamics in aDuffing-Van derPol oscillator with 1205956 potentialrdquo Journal of the Physical Societyof Japan vol 77 no 11 Article ID 114003 2008
[14] H G Patel and S N Sharma ldquoFiltering for a Duffing-van derPol stochastic differential equationrdquo Applied Mathematics andComputation vol 226 pp 386ndash397 2014
[15] C Li W Xu L Wang and D X Li ldquoStochastic responsesof Duffing-Van der Pol vibro-impact system under additivecolored noise excitationrdquo Chinese Physics B vol 22 no 11Article ID 110205 2013
[16] J B Li Chaos and Melnikov Method Chongqing UniversityPress Chongqing China 1989 (Chinese)
[17] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems andChaos vol 2 ofTexts in AppliedMathematics SpringerNew York NY USA 1990
[18] M S Siewe C Tchawoua and S Rajasekar ldquoHomoclinicbifurcation and chaos in 1206016-Rayleigh oscillator with three wellsdriven by an amplitude modulated forcerdquo International Journalof Bifurcation and Chaos vol 21 no 6 pp 1583ndash1593 2011
[19] V Ravichandran V Chinnathambi and S Rajasekar ldquoHomo-clinic bifurcation and chaos in Duffing oscillator driven by anamplitude-modulated forcerdquo Physica A Statistical Mechanicsand its Applications vol 376 pp 223ndash236 2007
[20] C Castilho and M Marchesin ldquoA practical use of the Melnikovhomoclinic methodrdquo Journal of Mathematical Physics vol 50no 11 Article ID 112704 11 pages 2009
[21] L Shi Y Zou and T Kupper ldquoMelnikov method and detectionof chaos for non-smooth systemsrdquo Acta Mathematicae Appli-catae Sinica English Series vol 29 no 4 pp 881ndash896 2013
