Research ArticleApplication of Extended Homotopy Analysis Method to theTwo-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator
Y H Qian S M Chen and L Shen
College of Mathematics Physics and Information Engineering Zhejiang Normal University Jinhua Zhejiang 321004 China
Correspondence should be addressed to Y H Qian qyh2004zjnucn
Received 19 January 2014 Accepted 21 February 2014 Published 30 March 2014
Academic Editor Jinlu Li
Copyright copy 2014 Y H Qian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator Meanwhile the comparisons between the results of the EHAM andstandard Runge-Kutta numerical method are also presented The results demonstrate that the analytical approximate solutions ofthe EHAM agree well with the numerical integration solutions For EHAM as an analytical approximation method we are not surewhether it can apply to all of the nonlinear systems we can only verify its effectiveness through specific cases As a result of theexistence of nonlinear terms we must study different types of systems no matter from the complication of calculation and physicalsignificance
1 Introduction
Mathematical methods for the natural and engineeringsciences problems have drawn considerable attention inrecent years In normal circumstances most of the nonlineardynamical models can be governed by a set of differentialequations and auxiliary conditions from modeling processes[1] Numerous analytical methods have been developed todeal with the nonlinear differential equations such as themodified perturbation methods [2ndash5] improved harmonicbalance methods [6 7] energy balance method [8 9] andthe frequency-amplitude formulation [10 11] Enlighteningfrom the basic concepts of the homotopy in topology [12 13]Liao developed the homotopy analysis method (HAM) [14ndash16] which does not require small parameters as one of theefficient analytical techniques in solving a variety of nonlinearvibration problems
Recently great attention is paid to the discussion ofcoupled oscillators of nonlinear dynamical systems becausemost of practical engineering problems can be governedby such coupled systems [17ndash19] The extended homotopyanalysis method (EHAM) is one method based on the HAMenvisioned first by Liao [16] More recently Qian et al[20] extended the HAM to deal with strongly nonlinear
coupled van der Pol oscillators For EHAM as an analyticalapproximation method we are not sure whether it canapply to all of the nonlinear systems we can only verifyits effectiveness through specific cases As a result of theexistence of nonlinear terms wemust study different types ofsystems no matter from the complication of calculation andphysical significance By solving such example it is illustratedthat the present techniques are not an adhoc approach itcan be generalized to investigate more complicated nonlinearmulti-degree-of-freedom (MDOF) dynamical systems
In the present work the exact analytical series solutionsof the two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing system are obtained by using the EHAM and we alsoestablish the comparisons between the results of the EHAMand standard Runge-Kutta numerical method It is shownthat the periodic solutions of the EHAM are in excellentagreement with the numerical integration ones even if time119905 progresses to a certain large domain In what followsSection 2 presents the EHAM of the MDOF dynamicalsystem Moreover the EHAM is presented to establish theanalytical approximate solutions for 2-DOF coupled vander Pol-Duffing oscillator in the next section In Section 4numerical comparisons are carried out to authenticate the
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 729184 7 pageshttpdxdoiorg1011552014729184
2 Abstract and Applied Analysis
correctness and accuracy of the present method Finally thepaper ends with concluding remarks in Section 5
2 The Extended Homotopy Analysis Method
TheMDOF dynamical system is considered by the followingequation
where 119902 is an 119899-dimensional unknown vector a dot denotesthe derivative with respect to time 119905 119872 119866 and 119870 arerespectively 119899times 119899mass damping and stiffness matrixes and119865 is the vector function of 119902 119902 and 119905 Let 119865( 119902 119902 119905) equiv 0 then(1) is an autonomous dynamical system
+ 119870119906 (119903 119905) minus 119865(120597119906
120597119905 119906 119905)
(2)
where 119906(119903 119905) is an unknown vector value function and 119903 and119905 are spatial and temporal variables respectively
In (2) the unknown vector functions of 119906(119903 119905)120597119906(119903 119905)120597119905 and 1205972119906(119903 119905)1205971199052 are respectively
119906 (119903 119905) = (1199091(119905) 119909
119899(119905))119879
120597119906 (119903 119905)
120597119905= (1198891199091
119889119905
119889119909119899
119889119905)
119879
1205972119906 (119903 119905)
1205971199052
= (11988921199091
1198891199052
1198892119909119899
1198891199052)
119879
(3)
According to the fundamental concepts and workingprocedures of the HAM [1 2] the zeroth-order deformationequation can be constructed as follows
(1 minus 119901) 119871 [Φ (119903 119905 119901) minus 1199060(119903 119905)]
and expandingΦ(119903 119905 119901) into the Taylor series expansionwithrespect to 119901 in accordance with the theorem of vector-valuedfunction we obtain
If the auxiliary linear operator initial guess solutionauxiliary parameters ℎ
119894 and auxiliary functions 119867
119894(119905) are
properly chosen the series equation (10) converges at 119901 = 1and we arrive at
119906 (119903 119905) = 1199060(119903 119905) +
+infin
sum
119898=1
119906119898(119903 119905) (11)
For brevity the vector of u119898is defined as
u119898= 1199060(119903 119905) 119906
1(119903 119905) 119906
119898(119903 119905) (12)
Differentiating the zeroth-order deformation equation(4) 119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 yield
The119898th-order deformation equation (13) is a linear equa-tion which can be readily solved by the symbolic softwaresuch as Mathematica
3 Application of the EHAM
In this section we apply the EHAM for analysis of the twocoupled van der Pol-Duffing oscillators
1+ 1205761205781(1199092
1minus 1)
1+ 1199091+ 12057612057211199093
1+ 120576120575111990911199092
2= 0 (15a)
2+ 1205761205782(1199092
2minus 1)
2+ 1199092+ 12057612057221199093
2+ 120576120575211990921199092
1= 0 (15b)
where the superscript denotes the differentiation with respectto time 119905 119909
1(119905) and 119909
2(119905) are the unknown real functions and
120576 1205721 1205722 1205781 1205782 1205751 and 120575
2are parameters
We introduce a new variable 120591 and substitute 120591 = 120596 1199051199091(119905) = 119906
1(120591) and 119909
2(119905) = 119906
2(120591) into (15a) and (15b)
Therefore we have
120596211990610158401015840
1+ 120596120576120578
1(1199062
1minus 1) 119906
1015840
1+ 1199061+ 12057612057211199063
1+ 120576120575111990611199062
2= 0 (16a)
120596211990610158401015840
2+ 120596120576120578
2(1199062
2minus 1) 119906
1015840
2+ 1199062+ 12057612057221199063
2+ 120576120575211990621199062
1= 0
(16b)
subject to the initial conditions
1199061(0) = 119886 119906
1015840
1(0) = 119887 119906
2(0) = 119888 119906
1015840
2(0) = 0 (17)
where a prime denotes the derivative with respect to variable120591 Provided that the periodic solutions in (16a) and (16b) canbe expressed by a set of base functions
cos (119896120591) sin (119896120591) | 119896 = 0 1 2 (18)
one obtains
1199061(120591) =
+infin
sum
119896=0
(1198861119896
cos 119896120591 + 1198871119896
sin 119896120591) (19a)
1199062(120591) =
+infin
sum
119896=0
(1198862119896
cos 119896120591 + 1198872119896
sin 119896120591) (19b)
For the initial approximation 11990610(120591) and 119906
20(120591) are
assumed as
11990610(120591) = 119886
0cos 120591 + 119887
0sin 120591 119906
20(120591) = 119888
0cos 120591 (20)
and the linear operator is defined as
119871(Φ1(120591 119901)
Φ2(120591 119901)
) = 1205962
0(
1205972Φ1(120591 119901)
1205971205912
+ Φ1(120591 119901)
1205972Φ2(120591 119901)
1205971205912
+ Φ2(120591 119901)
) (21)
We can define a nonlinear operator as the following byEHAM
119873(Φ1(120591 119901)
Φ2(120591 119901)
) =
(((
(
Ω2(119901)1205972Φ1(120591 119901)
1205971205912
+ 1205761205781Ω(119901) [Φ
1
2(120591 119901) minus 1]
120597Φ1(120591 119901)
120597120591
+Φ1(120591 119901) + 120576120572
1Φ1
3(120591 119901) + 120576120575
1Φ1(120591 119901)Φ
2
2(120591 119901)
Ω2(119901)1205972Φ2(120591 119901)
1205971205912
+ 1205761205782Ω(119901) [Φ
2
2(120591 119901) minus 1]
120597Φ2(120591 119901)
120597120591
+Φ2(120591 119901) + 120576120572
2Φ2
3(120591 119901) + 120576120575
2Φ1
2(120591 119901)Φ
2(120591 119901)
)))
)
(22)
and the nonlinear operator Π is
Π(Φ1(120591 119901)
Φ2(120591 119901)
) = (1 minus 119901)
times [(119860 (119901) cos 120591 + 119861 (119901) sin 120591
119862 (119901) cos 120591 )
+((Ω2(119901) minus 120596
2
0) (cos 120591 + sin 120591)
(Ω2(119901) minus 120596
2
0) sin 120591
)]
(23)
In terms of the principle of solution expression we selectthe auxiliary functions as 119867
1(119905) = 1 and 119867
2(119905) = 1 thus the
zeroth-order deformation equation is given by
(1 minus 119901) 119871(
Φ1(120591 119901) minus 119906
10(120591)
Φ2(120591 119901) minus 119906
20(120591)
)
= 119901ℎ1119873(
Φ1(120591 119901)
Φ2(120591 119901)
) + (1 minus 119901)
4 Abstract and Applied Analysis
times ℎ2[(119860 (119901) cos 120591 + 119861 (119901) sin 120591
If the auxiliary parameters ℎ1and ℎ2are properly chosen
the power series solutions in (29a) (29b) and (29c) areconverged at 119901 = 1 Then from (28) we get
1199061(120591) = 119906
10(120591) +
+infin
sum
119898=1
1199061119898(120591)
1199062(120591) = 119906
20(120591) +
+infin
sum
119898=1
1199062119898(120591)
120596 = 1205960+
+infin
sum
119898=1
120596119898
(31)
For simplicity the following vectors are defined as
rarr
1199061119899= 11990610(120591) 119906
11(120591) 119906
1119899(120591) (32a)
rarr
1199062119899= 11990620(120591) 119906
21(120591) 119906
2119899(120591) (32b)
rarr
120596119899= 1205960 1205961 120596
119899 (32c)
By differentiating the zeroth-order deformation equation(24)119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 the119898th-order deformation equation isformulated as follows
Because of the principle of solution expression and thelinear operator 119871 the right side of (33) should not containthe terms of sin 120591 and cos 120591 or the secular terms 120591 sin 120591 and120591 cos 120591 The coefficients are set to be zero to yield
1
120587int
2120587
0
[ℎ11198771119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36a)
1
120587int
2120587
0
[ℎ11198771 119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36b)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36c)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36d)
The solutions of 120596119896 119886119896 119887119896 and 119888
119896(119896 = 0 1 2 )
from (33) and (36a) (36b) (36c) and (36d) can be computedsuccessively To achieve more accurate results we modify thesolution of 120596 as follows
120596 = 120596 + 120592 (37)
where 120592 is a small parameter
4 Numerical Simulation and Discussion
In this section numerical experiment is conducted to verifythe accuracy of the present approach
Taking 120576 = 1100 1205781= 1 120578
2= 2 120572
1= 12 120572
2= 13
1205751= 210 120575
2= 310 and the initial approximations of 119886 119887
119888 and 120596 are 1198860= 2 119887
0= 0 119888
0= 2 and 120596
0= 10871000
respectivelyFor simplicity and accuracy we set ℎ
1= minus01 ℎ
2= 300
and 120592 = minus01 then the comparison of the phase curves ofthe fifth-order approximation with the numerical integrationsolution is shown in Figure 1
6 Abstract and Applied Analysis
0
Approximation solutionNumerical solution
1 2
0
1
2x998400 1
x1
minus1
minus1
minus2
minus2
(a)
0
Approximation solutionNumerical solution
1 2
0
1
2
x998400 2
x2
minus1
minus1
minus2
minus2
(b)
Figure 2 Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution
The initial conditions of the numerical integrationmethod are 119909
1(0) = 1999538 1199091015840
1(0) = 0000037 119909
2(0) =
1999458 and 11990910158402(0) = 0
Moreover the fifth-order analytical solutions (1199091(119905)
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
where 119902 is an 119899-dimensional unknown vector a dot denotesthe derivative with respect to time 119905 119872 119866 and 119870 arerespectively 119899times 119899mass damping and stiffness matrixes and119865 is the vector function of 119902 119902 and 119905 Let 119865( 119902 119902 119905) equiv 0 then(1) is an autonomous dynamical system
+ 119870119906 (119903 119905) minus 119865(120597119906
120597119905 119906 119905)
(2)
where 119906(119903 119905) is an unknown vector value function and 119903 and119905 are spatial and temporal variables respectively
In (2) the unknown vector functions of 119906(119903 119905)120597119906(119903 119905)120597119905 and 1205972119906(119903 119905)1205971199052 are respectively
119906 (119903 119905) = (1199091(119905) 119909
119899(119905))119879
120597119906 (119903 119905)
120597119905= (1198891199091
119889119905
119889119909119899
119889119905)
119879
1205972119906 (119903 119905)
1205971199052
= (11988921199091
1198891199052
1198892119909119899
1198891199052)
119879
(3)
According to the fundamental concepts and workingprocedures of the HAM [1 2] the zeroth-order deformationequation can be constructed as follows
(1 minus 119901) 119871 [Φ (119903 119905 119901) minus 1199060(119903 119905)]
and expandingΦ(119903 119905 119901) into the Taylor series expansionwithrespect to 119901 in accordance with the theorem of vector-valuedfunction we obtain
If the auxiliary linear operator initial guess solutionauxiliary parameters ℎ
119894 and auxiliary functions 119867
119894(119905) are
properly chosen the series equation (10) converges at 119901 = 1and we arrive at
119906 (119903 119905) = 1199060(119903 119905) +
+infin
sum
119898=1
119906119898(119903 119905) (11)
For brevity the vector of u119898is defined as
u119898= 1199060(119903 119905) 119906
1(119903 119905) 119906
119898(119903 119905) (12)
Differentiating the zeroth-order deformation equation(4) 119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 yield
The119898th-order deformation equation (13) is a linear equa-tion which can be readily solved by the symbolic softwaresuch as Mathematica
3 Application of the EHAM
In this section we apply the EHAM for analysis of the twocoupled van der Pol-Duffing oscillators
1+ 1205761205781(1199092
1minus 1)
1+ 1199091+ 12057612057211199093
1+ 120576120575111990911199092
2= 0 (15a)
2+ 1205761205782(1199092
2minus 1)
2+ 1199092+ 12057612057221199093
2+ 120576120575211990921199092
1= 0 (15b)
where the superscript denotes the differentiation with respectto time 119905 119909
1(119905) and 119909
2(119905) are the unknown real functions and
120576 1205721 1205722 1205781 1205782 1205751 and 120575
2are parameters
We introduce a new variable 120591 and substitute 120591 = 120596 1199051199091(119905) = 119906
1(120591) and 119909
2(119905) = 119906
2(120591) into (15a) and (15b)
Therefore we have
120596211990610158401015840
1+ 120596120576120578
1(1199062
1minus 1) 119906
1015840
1+ 1199061+ 12057612057211199063
1+ 120576120575111990611199062
2= 0 (16a)
120596211990610158401015840
2+ 120596120576120578
2(1199062
2minus 1) 119906
1015840
2+ 1199062+ 12057612057221199063
2+ 120576120575211990621199062
1= 0
(16b)
subject to the initial conditions
1199061(0) = 119886 119906
1015840
1(0) = 119887 119906
2(0) = 119888 119906
1015840
2(0) = 0 (17)
where a prime denotes the derivative with respect to variable120591 Provided that the periodic solutions in (16a) and (16b) canbe expressed by a set of base functions
cos (119896120591) sin (119896120591) | 119896 = 0 1 2 (18)
one obtains
1199061(120591) =
+infin
sum
119896=0
(1198861119896
cos 119896120591 + 1198871119896
sin 119896120591) (19a)
1199062(120591) =
+infin
sum
119896=0
(1198862119896
cos 119896120591 + 1198872119896
sin 119896120591) (19b)
For the initial approximation 11990610(120591) and 119906
20(120591) are
assumed as
11990610(120591) = 119886
0cos 120591 + 119887
0sin 120591 119906
20(120591) = 119888
0cos 120591 (20)
and the linear operator is defined as
119871(Φ1(120591 119901)
Φ2(120591 119901)
) = 1205962
0(
1205972Φ1(120591 119901)
1205971205912
+ Φ1(120591 119901)
1205972Φ2(120591 119901)
1205971205912
+ Φ2(120591 119901)
) (21)
We can define a nonlinear operator as the following byEHAM
119873(Φ1(120591 119901)
Φ2(120591 119901)
) =
(((
(
Ω2(119901)1205972Φ1(120591 119901)
1205971205912
+ 1205761205781Ω(119901) [Φ
1
2(120591 119901) minus 1]
120597Φ1(120591 119901)
120597120591
+Φ1(120591 119901) + 120576120572
1Φ1
3(120591 119901) + 120576120575
1Φ1(120591 119901)Φ
2
2(120591 119901)
Ω2(119901)1205972Φ2(120591 119901)
1205971205912
+ 1205761205782Ω(119901) [Φ
2
2(120591 119901) minus 1]
120597Φ2(120591 119901)
120597120591
+Φ2(120591 119901) + 120576120572
2Φ2
3(120591 119901) + 120576120575
2Φ1
2(120591 119901)Φ
2(120591 119901)
)))
)
(22)
and the nonlinear operator Π is
Π(Φ1(120591 119901)
Φ2(120591 119901)
) = (1 minus 119901)
times [(119860 (119901) cos 120591 + 119861 (119901) sin 120591
119862 (119901) cos 120591 )
+((Ω2(119901) minus 120596
2
0) (cos 120591 + sin 120591)
(Ω2(119901) minus 120596
2
0) sin 120591
)]
(23)
In terms of the principle of solution expression we selectthe auxiliary functions as 119867
1(119905) = 1 and 119867
2(119905) = 1 thus the
zeroth-order deformation equation is given by
(1 minus 119901) 119871(
Φ1(120591 119901) minus 119906
10(120591)
Φ2(120591 119901) minus 119906
20(120591)
)
= 119901ℎ1119873(
Φ1(120591 119901)
Φ2(120591 119901)
) + (1 minus 119901)
4 Abstract and Applied Analysis
times ℎ2[(119860 (119901) cos 120591 + 119861 (119901) sin 120591
If the auxiliary parameters ℎ1and ℎ2are properly chosen
the power series solutions in (29a) (29b) and (29c) areconverged at 119901 = 1 Then from (28) we get
1199061(120591) = 119906
10(120591) +
+infin
sum
119898=1
1199061119898(120591)
1199062(120591) = 119906
20(120591) +
+infin
sum
119898=1
1199062119898(120591)
120596 = 1205960+
+infin
sum
119898=1
120596119898
(31)
For simplicity the following vectors are defined as
rarr
1199061119899= 11990610(120591) 119906
11(120591) 119906
1119899(120591) (32a)
rarr
1199062119899= 11990620(120591) 119906
21(120591) 119906
2119899(120591) (32b)
rarr
120596119899= 1205960 1205961 120596
119899 (32c)
By differentiating the zeroth-order deformation equation(24)119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 the119898th-order deformation equation isformulated as follows
Because of the principle of solution expression and thelinear operator 119871 the right side of (33) should not containthe terms of sin 120591 and cos 120591 or the secular terms 120591 sin 120591 and120591 cos 120591 The coefficients are set to be zero to yield
1
120587int
2120587
0
[ℎ11198771119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36a)
1
120587int
2120587
0
[ℎ11198771 119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36b)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36c)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36d)
The solutions of 120596119896 119886119896 119887119896 and 119888
119896(119896 = 0 1 2 )
from (33) and (36a) (36b) (36c) and (36d) can be computedsuccessively To achieve more accurate results we modify thesolution of 120596 as follows
120596 = 120596 + 120592 (37)
where 120592 is a small parameter
4 Numerical Simulation and Discussion
In this section numerical experiment is conducted to verifythe accuracy of the present approach
Taking 120576 = 1100 1205781= 1 120578
2= 2 120572
1= 12 120572
2= 13
1205751= 210 120575
2= 310 and the initial approximations of 119886 119887
119888 and 120596 are 1198860= 2 119887
0= 0 119888
0= 2 and 120596
0= 10871000
respectivelyFor simplicity and accuracy we set ℎ
1= minus01 ℎ
2= 300
and 120592 = minus01 then the comparison of the phase curves ofthe fifth-order approximation with the numerical integrationsolution is shown in Figure 1
6 Abstract and Applied Analysis
0
Approximation solutionNumerical solution
1 2
0
1
2x998400 1
x1
minus1
minus1
minus2
minus2
(a)
0
Approximation solutionNumerical solution
1 2
0
1
2
x998400 2
x2
minus1
minus1
minus2
minus2
(b)
Figure 2 Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution
The initial conditions of the numerical integrationmethod are 119909
1(0) = 1999538 1199091015840
1(0) = 0000037 119909
2(0) =
1999458 and 11990910158402(0) = 0
Moreover the fifth-order analytical solutions (1199091(119905)
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
The119898th-order deformation equation (13) is a linear equa-tion which can be readily solved by the symbolic softwaresuch as Mathematica
3 Application of the EHAM
In this section we apply the EHAM for analysis of the twocoupled van der Pol-Duffing oscillators
1+ 1205761205781(1199092
1minus 1)
1+ 1199091+ 12057612057211199093
1+ 120576120575111990911199092
2= 0 (15a)
2+ 1205761205782(1199092
2minus 1)
2+ 1199092+ 12057612057221199093
2+ 120576120575211990921199092
1= 0 (15b)
where the superscript denotes the differentiation with respectto time 119905 119909
1(119905) and 119909
2(119905) are the unknown real functions and
120576 1205721 1205722 1205781 1205782 1205751 and 120575
2are parameters
We introduce a new variable 120591 and substitute 120591 = 120596 1199051199091(119905) = 119906
1(120591) and 119909
2(119905) = 119906
2(120591) into (15a) and (15b)
Therefore we have
120596211990610158401015840
1+ 120596120576120578
1(1199062
1minus 1) 119906
1015840
1+ 1199061+ 12057612057211199063
1+ 120576120575111990611199062
2= 0 (16a)
120596211990610158401015840
2+ 120596120576120578
2(1199062
2minus 1) 119906
1015840
2+ 1199062+ 12057612057221199063
2+ 120576120575211990621199062
1= 0
(16b)
subject to the initial conditions
1199061(0) = 119886 119906
1015840
1(0) = 119887 119906
2(0) = 119888 119906
1015840
2(0) = 0 (17)
where a prime denotes the derivative with respect to variable120591 Provided that the periodic solutions in (16a) and (16b) canbe expressed by a set of base functions
cos (119896120591) sin (119896120591) | 119896 = 0 1 2 (18)
one obtains
1199061(120591) =
+infin
sum
119896=0
(1198861119896
cos 119896120591 + 1198871119896
sin 119896120591) (19a)
1199062(120591) =
+infin
sum
119896=0
(1198862119896
cos 119896120591 + 1198872119896
sin 119896120591) (19b)
For the initial approximation 11990610(120591) and 119906
20(120591) are
assumed as
11990610(120591) = 119886
0cos 120591 + 119887
0sin 120591 119906
20(120591) = 119888
0cos 120591 (20)
and the linear operator is defined as
119871(Φ1(120591 119901)
Φ2(120591 119901)
) = 1205962
0(
1205972Φ1(120591 119901)
1205971205912
+ Φ1(120591 119901)
1205972Φ2(120591 119901)
1205971205912
+ Φ2(120591 119901)
) (21)
We can define a nonlinear operator as the following byEHAM
119873(Φ1(120591 119901)
Φ2(120591 119901)
) =
(((
(
Ω2(119901)1205972Φ1(120591 119901)
1205971205912
+ 1205761205781Ω(119901) [Φ
1
2(120591 119901) minus 1]
120597Φ1(120591 119901)
120597120591
+Φ1(120591 119901) + 120576120572
1Φ1
3(120591 119901) + 120576120575
1Φ1(120591 119901)Φ
2
2(120591 119901)
Ω2(119901)1205972Φ2(120591 119901)
1205971205912
+ 1205761205782Ω(119901) [Φ
2
2(120591 119901) minus 1]
120597Φ2(120591 119901)
120597120591
+Φ2(120591 119901) + 120576120572
2Φ2
3(120591 119901) + 120576120575
2Φ1
2(120591 119901)Φ
2(120591 119901)
)))
)
(22)
and the nonlinear operator Π is
Π(Φ1(120591 119901)
Φ2(120591 119901)
) = (1 minus 119901)
times [(119860 (119901) cos 120591 + 119861 (119901) sin 120591
119862 (119901) cos 120591 )
+((Ω2(119901) minus 120596
2
0) (cos 120591 + sin 120591)
(Ω2(119901) minus 120596
2
0) sin 120591
)]
(23)
In terms of the principle of solution expression we selectthe auxiliary functions as 119867
1(119905) = 1 and 119867
2(119905) = 1 thus the
zeroth-order deformation equation is given by
(1 minus 119901) 119871(
Φ1(120591 119901) minus 119906
10(120591)
Φ2(120591 119901) minus 119906
20(120591)
)
= 119901ℎ1119873(
Φ1(120591 119901)
Φ2(120591 119901)
) + (1 minus 119901)
4 Abstract and Applied Analysis
times ℎ2[(119860 (119901) cos 120591 + 119861 (119901) sin 120591
If the auxiliary parameters ℎ1and ℎ2are properly chosen
the power series solutions in (29a) (29b) and (29c) areconverged at 119901 = 1 Then from (28) we get
1199061(120591) = 119906
10(120591) +
+infin
sum
119898=1
1199061119898(120591)
1199062(120591) = 119906
20(120591) +
+infin
sum
119898=1
1199062119898(120591)
120596 = 1205960+
+infin
sum
119898=1
120596119898
(31)
For simplicity the following vectors are defined as
rarr
1199061119899= 11990610(120591) 119906
11(120591) 119906
1119899(120591) (32a)
rarr
1199062119899= 11990620(120591) 119906
21(120591) 119906
2119899(120591) (32b)
rarr
120596119899= 1205960 1205961 120596
119899 (32c)
By differentiating the zeroth-order deformation equation(24)119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 the119898th-order deformation equation isformulated as follows
Because of the principle of solution expression and thelinear operator 119871 the right side of (33) should not containthe terms of sin 120591 and cos 120591 or the secular terms 120591 sin 120591 and120591 cos 120591 The coefficients are set to be zero to yield
1
120587int
2120587
0
[ℎ11198771119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36a)
1
120587int
2120587
0
[ℎ11198771 119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36b)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36c)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36d)
The solutions of 120596119896 119886119896 119887119896 and 119888
119896(119896 = 0 1 2 )
from (33) and (36a) (36b) (36c) and (36d) can be computedsuccessively To achieve more accurate results we modify thesolution of 120596 as follows
120596 = 120596 + 120592 (37)
where 120592 is a small parameter
4 Numerical Simulation and Discussion
In this section numerical experiment is conducted to verifythe accuracy of the present approach
Taking 120576 = 1100 1205781= 1 120578
2= 2 120572
1= 12 120572
2= 13
1205751= 210 120575
2= 310 and the initial approximations of 119886 119887
119888 and 120596 are 1198860= 2 119887
0= 0 119888
0= 2 and 120596
0= 10871000
respectivelyFor simplicity and accuracy we set ℎ
1= minus01 ℎ
2= 300
and 120592 = minus01 then the comparison of the phase curves ofthe fifth-order approximation with the numerical integrationsolution is shown in Figure 1
6 Abstract and Applied Analysis
0
Approximation solutionNumerical solution
1 2
0
1
2x998400 1
x1
minus1
minus1
minus2
minus2
(a)
0
Approximation solutionNumerical solution
1 2
0
1
2
x998400 2
x2
minus1
minus1
minus2
minus2
(b)
Figure 2 Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution
The initial conditions of the numerical integrationmethod are 119909
1(0) = 1999538 1199091015840
1(0) = 0000037 119909
2(0) =
1999458 and 11990910158402(0) = 0
Moreover the fifth-order analytical solutions (1199091(119905)
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
If the auxiliary parameters ℎ1and ℎ2are properly chosen
the power series solutions in (29a) (29b) and (29c) areconverged at 119901 = 1 Then from (28) we get
1199061(120591) = 119906
10(120591) +
+infin
sum
119898=1
1199061119898(120591)
1199062(120591) = 119906
20(120591) +
+infin
sum
119898=1
1199062119898(120591)
120596 = 1205960+
+infin
sum
119898=1
120596119898
(31)
For simplicity the following vectors are defined as
rarr
1199061119899= 11990610(120591) 119906
11(120591) 119906
1119899(120591) (32a)
rarr
1199062119899= 11990620(120591) 119906
21(120591) 119906
2119899(120591) (32b)
rarr
120596119899= 1205960 1205961 120596
119899 (32c)
By differentiating the zeroth-order deformation equation(24)119898 times with respect to 119901 then dividing the equation by119898 and setting 119901 = 0 the119898th-order deformation equation isformulated as follows
Because of the principle of solution expression and thelinear operator 119871 the right side of (33) should not containthe terms of sin 120591 and cos 120591 or the secular terms 120591 sin 120591 and120591 cos 120591 The coefficients are set to be zero to yield
1
120587int
2120587
0
[ℎ11198771119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36a)
1
120587int
2120587
0
[ℎ11198771 119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36b)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36c)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36d)
The solutions of 120596119896 119886119896 119887119896 and 119888
119896(119896 = 0 1 2 )
from (33) and (36a) (36b) (36c) and (36d) can be computedsuccessively To achieve more accurate results we modify thesolution of 120596 as follows
120596 = 120596 + 120592 (37)
where 120592 is a small parameter
4 Numerical Simulation and Discussion
In this section numerical experiment is conducted to verifythe accuracy of the present approach
Taking 120576 = 1100 1205781= 1 120578
2= 2 120572
1= 12 120572
2= 13
1205751= 210 120575
2= 310 and the initial approximations of 119886 119887
119888 and 120596 are 1198860= 2 119887
0= 0 119888
0= 2 and 120596
0= 10871000
respectivelyFor simplicity and accuracy we set ℎ
1= minus01 ℎ
2= 300
and 120592 = minus01 then the comparison of the phase curves ofthe fifth-order approximation with the numerical integrationsolution is shown in Figure 1
6 Abstract and Applied Analysis
0
Approximation solutionNumerical solution
1 2
0
1
2x998400 1
x1
minus1
minus1
minus2
minus2
(a)
0
Approximation solutionNumerical solution
1 2
0
1
2
x998400 2
x2
minus1
minus1
minus2
minus2
(b)
Figure 2 Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution
The initial conditions of the numerical integrationmethod are 119909
1(0) = 1999538 1199091015840
1(0) = 0000037 119909
2(0) =
1999458 and 11990910158402(0) = 0
Moreover the fifth-order analytical solutions (1199091(119905)
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
Because of the principle of solution expression and thelinear operator 119871 the right side of (33) should not containthe terms of sin 120591 and cos 120591 or the secular terms 120591 sin 120591 and120591 cos 120591 The coefficients are set to be zero to yield
1
120587int
2120587
0
[ℎ11198771119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36a)
1
120587int
2120587
0
[ℎ11198771 119898(rarr
1199061 119898minus1
rarr
120596119898minus1)
+ℎ21198781119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36b)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] cos 120591119889120591 = 0
(36c)
1
120587int
2120587
0
[ℎ11198772 119898(rarr
1199062 119898minus1
rarr
120596119898minus1)
+ℎ21198782119898(120591rarr
120596119898)] sin 120591119889120591 = 0
(36d)
The solutions of 120596119896 119886119896 119887119896 and 119888
119896(119896 = 0 1 2 )
from (33) and (36a) (36b) (36c) and (36d) can be computedsuccessively To achieve more accurate results we modify thesolution of 120596 as follows
120596 = 120596 + 120592 (37)
where 120592 is a small parameter
4 Numerical Simulation and Discussion
In this section numerical experiment is conducted to verifythe accuracy of the present approach
Taking 120576 = 1100 1205781= 1 120578
2= 2 120572
1= 12 120572
2= 13
1205751= 210 120575
2= 310 and the initial approximations of 119886 119887
119888 and 120596 are 1198860= 2 119887
0= 0 119888
0= 2 and 120596
0= 10871000
respectivelyFor simplicity and accuracy we set ℎ
1= minus01 ℎ
2= 300
and 120592 = minus01 then the comparison of the phase curves ofthe fifth-order approximation with the numerical integrationsolution is shown in Figure 1
6 Abstract and Applied Analysis
0
Approximation solutionNumerical solution
1 2
0
1
2x998400 1
x1
minus1
minus1
minus2
minus2
(a)
0
Approximation solutionNumerical solution
1 2
0
1
2
x998400 2
x2
minus1
minus1
minus2
minus2
(b)
Figure 2 Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution
The initial conditions of the numerical integrationmethod are 119909
1(0) = 1999538 1199091015840
1(0) = 0000037 119909
2(0) =
1999458 and 11990910158402(0) = 0
Moreover the fifth-order analytical solutions (1199091(119905)
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
+ 000068900 cos 3120596119905 minus 000106809 sin 3120596119905
1199092(119905) = minus199883388 cos120596119905 minus 000641395 sin120596119905
minus 000062613 cos 3120596t minus 000213658 sin 3120596119905
120596 = 108703369
(40)
in which
120596 = 120596 + 120592 = 098703369 (41)
5 Conclusions
In the present paper the EHAM approach is applied to getasymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling The basic ideadescribed in this paper is expected to be more employedin solving other dynamical systems in engineering andphysical sciences Comparisons with the numerical resultsare presented to demonstrate the validity of this method Insummary compared with some other methods the EHAMhas the following advantages
(1) The EHAM provides an ingenious avenue for con-trolling the convergences of approximation seriesNumerical comparisons demonstrate that the EHAMis an effective and robust analytical method of 2-DOFvan der Pol-Duffing oscillators
Abstract and Applied Analysis 7
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
(2) Because of its flexibility the present techniques canalso be further generalized to analyze more compli-cated nonlinear MDOF dynamical systems that canonly be analyzed by numerical methods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
All the authors contributed equally and significantly to thewriting of this paper All the authors read and approved thefinal paper
Acknowledgments
The author Y H Qian gratefully acknowledges the sup-port of the National Natural Science Foundations of China(NNSFC) throughGrants nos 11202189 and 11304286 and theNatural Science Foundation of Zhejiang Province of Chinathrough Grant no LY12A02002 The author S M Chengratefully appreciates the financial support from the NNSFCthrough Grants no 11371326 The author L Shen gratefullyacknowledges the support from open experiment project ofZhejiang Normal University The authors are also grateful tothe anonymous reviewers for their constructive commentsand suggestions
References
[1] R E Mickens Mathematical Methods for the Natural andEngineering Sciences World Scientific Singapore 2004
[2] Y K Cheung S H Chen and S L Lau ldquoAmodified Lindstedt-Poincare method for certain strongly nonlinear oscillatorsrdquoInternational Journal of Non-Linear Mechanics vol 26 no 3-4pp 367ndash378 1991
[3] M Senator and C N Bapat ldquoA perturbation technique thatworks evenwhen the nonlinearity is not smallrdquo Journal of Soundand Vibration vol 164 no 1 pp 1ndash27 1993
[4] P Amore and A Aranda ldquoImproved Lindstedt-Poincaremethod for the solution of nonlinear problemsrdquo Journal ofSound and Vibration vol 283 no 3ndash5 pp 1115ndash1136 2005
[5] R R Pusenjak ldquoExtended Lindstedt-Poincare method fornon-stationary resonances of dynamical systems with cubicnonlinearitiesrdquo Journal of Sound and Vibration vol 314 no 1-2 pp 194ndash216 2008
[6] J L Summers and M D Savage ldquoTwo timescale harmonicbalance I Application to autonomous one-dimensional non-linear oscillatorsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 340 no1659 pp 473ndash501 1992
[7] B Wu and P Li ldquoA method for obtaining approximate analyticperiods for a class of nonlinear oscillatorsrdquo Meccanica vol 36no 2 pp 167ndash176 2001
[8] S S Ganji D D Ganji Z Z Ganji and S Karimpour ldquoPeriodicsolution for strongly nonlinear vibration systems byHersquos energybalance methodrdquo Acta Applicandae Mathematicae vol 106 no1 pp 79ndash92 2009
[9] I Mehdipour D D Ganji and M Mozaffari ldquoApplication ofthe energy balance method to nonlinear vibrating equationsrdquoCurrent Applied Physics vol 10 no 1 pp 104ndash112 2010
[10] L Zhao ldquoHersquos frequency-amplitude formulation for nonlinearoscillators with an irrational forcerdquo Computers amp Mathematicswith Applications vol 58 no 11-12 pp 2477ndash2479 2009
[11] J Fan ldquoHersquos frequency-amplitude formulation for the Duffingharmonic oscillatorrdquo Computers amp Mathematics with Applica-tions vol 58 no 11-12 pp 2473ndash2476 2009
[12] P J Hilton An Introduction to Homotopy Theory CambridgeUniversity Press Cambridge UK 1953
[13] C Nash and S Sen Topology and Geometry for PhysicistsAcademic Press London UK 1983
[14] S J Liao The proposed homotopy analysis techniques for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity Shanghai China 1992
[15] S J Liao ldquoA kind of approximate solution technique which doesnot depend upon small parameters II An application in fluidmechanicsrdquo International Journal of Non-Linear Mechanics vol32 no 5 pp 815ndash822 1997
[16] S Liao Beyond Perturbation Introduction to the HomotopyAnalysisMethod ChapmanampHall Boca Raton Fla USA 2004
[17] S Rajasekar and K Murali ldquoResonance behaviour and jumpphenomenon in a two coupled Duffing-van der Pol oscillatorsrdquoChaos Solitons and Fractals vol 19 no 4 pp 925ndash934 2004
[18] B T Nohara and A Arimoto ldquoNon-existence theorem exceptthe out-of-phase and in-phase solutions in the coupled van derPol equation systemrdquo Ukrainian Mathematical Journal vol 61no 8 pp 1311ndash1337 2009
[19] Y J Li B T Nohara and S J Liao ldquoSeries solutions of coupledvan der Pol equation by means of homotopy analysis methodrdquoJournal of Mathematical Physics vol 51 no 6 pp 1ndash12 2010
[20] YHQian CMDuan SM Chen and S P Chen ldquoAsymptoticanalytical solutions of the two-degree-of-freedom stronglynonlinear van der Pol oscillators with cubic couple terms usingextended homotopy analysismethodrdquoActaMechanica vol 223no 2 pp 237ndash255 2012
