A study of homotopy analysis method for limit cycle of van der Pol equation

Communications in Nonlinear Science and Numerical Simulation

Volumn 14, Issue 5, 2009, Pages 1816-1821

  Chen, Y M ^a   Liu, J K ^a  

^a SUN YAT SEN UNIVERSITY   (China)

Author keywords

Computational effort; Homotopy analysis method; Limit cycle; van der Pol equation

Indexed keywords

COMPUTATIONAL METHODS;

ANALYSIS METHOD; COMPUTATIONAL EFFORT; DEFORMATION EQUATIONS; HOMOTOPY ANALYSE METHOD; HOMOTOPY ANALYSIS; INITIAL CONDITIONS; LIMIT-CYCLE; POL EQUATION; VAN DER POL; VAN DER POL EQUATION;

NUMERICAL ANALYSIS;

EID: 56049101374     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2008.07.010     Document Type: Article

References (20)

1. Andronov A.A., Witt A.A., and Khaikin S.E. Theory of oscillations (1966), Pergamon Press, Oxford [1959 (English translation)]
2. Nayfeh A.H., and Mook D.T. Nonlinear oscillations (1979), Wiley, New York
3. Andersen C.M., and Geer J.F. Power series expansions for the frequency and period of the limit cycle of the van der Pol equation. SIAM J Appl Math 42 (1982) 678-693
4. Dafear M.B., Geer J., and Andersen C.M. Perturbation analysis of the limit cycle of the free van der Pol equation. SIAM J Appl Math 44 5 (1984) 881-895
5. Buonomo A. On the periodic solution of the van der Pol equation for small values of the damping parameter. Int J Circuit Theory Appl 26 (1998) 39-52
6. Chen Y.M., and Liu J.K. A modified Mickens iteration procedure for nonlinear oscillators. J Sound Vib 314 (2008) 465-473
7. Delamotte B. Nonperturbative (but approximate) method for solving differential equations and finding limit cycles. Phys Rev Lett 70 22 (1993) 3361-3364
8. Allgower E.L., and Georg K. Numerical continuation methods (1990), Springer-Verlag, Berlin
9. Watson L.T. Globally convergent homotopy algorithms for nonlinear systems of equations. Nonlinear Dyn 1 (1990) 143-191
10. Liao S.J. A second-order approximate analytical solution of a simple pendulum by the process analysis method. ASME J Appl Mech 59 (1992) 970-975
11. Liao S.J. An approximate solution technique not depending on small parameters: a special example. Int J Non-linear Mech 30 (1995) 371-380
12. Liao S.J. A kind of approximate solution technique which does not depend upon small parameters-II: an application in fluid mechanics. Int J Non-linear Mech 32 5 (1997) 815-822
13. Liao S.J. An explicit analytic solution to the Thomas-Fermi equation. Appl Math Comput 144 (2003) 495-506
14. Li S.C., and Liao S.J. An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl Math Comput 169 (2005) 854-865
15. Yang C., and Liao S.J. On the explicit, purely analytic solution of Von Kármán swirling viscous flow. Commun Nonlinear Sci Numer Simulat 11 (2006) 83-93
16. Liao S.J. An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Commun Nonlinear Sci Numer Simulat 11 (2006) 326-339
17. Xu H., and Liao S.J. Series solutions of unsteady MHD flows above a rotating disk. Acta Mech 41 6 (2006) 599-609
18. Tan Y., and Abbasbandy S. Homotopy analysis method for quadratic Riccati differential equation. Commun Nonlinear Sci Numer Simulat 13 (2008) 539-546
19. Liao S.J. On the homotopy analysis method for nonlinear problems. Appl Math Comput 147 (2004) 499-513
20. Liao S.J. An analytic approximate approach for free oscillations of self-excited system. Int J Non-linear Mech 39 (2004) 271-280