Skip to main content
Log in

Homoclinic Connections in Strongly Self-Excited Nonlinear Oscillators: The Melnikov Function and the Elliptic Lindstedt–Poincaré Method

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A criterion to predict bifurcation of homoclinic orbits instrongly nonlinear self-excited one-degree-of-freedom oscillator

$$\ddot x + c_1 x + c_2 f(x) = \varepsilon g(\mu ,x,\dot x),$$

is presented. TheLindstedt–Poincaré perturbation method is combined formally withthe Jacobian elliptic functions to determine an approximation of thelimit cycles near homoclinicity. We then apply a criterion forpredicting homoclinic orbits, based on the collision of the bifurcatinglimit cycle with the saddle equilibrium. In particular we show that thiscriterion leads to the same results, formally and to leading order, asthe standard Melnikov technique. Explicit applications of this criterionto quadratic or cubic nonlinearities f(x) are included.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, John Wiley, New York, 1979.

    Google Scholar 

  2. Jordan, D. W. and Smith, P., Nonlinear Ordinary Differential Equations, Oxford University Press, Oxford, 1987.

    Google Scholar 

  3. Nayfeh, A. H., Perturbation Methods, John Wiley, New York, 1973.

    Google Scholar 

  4. Nayfeh, A. H., Introduction to Perturbation Techniques, John Wiley, New York, 1981.

    Google Scholar 

  5. Krylov, N. and Bogolioubov, N., Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, NJ, 1943.

    Google Scholar 

  6. Bogolioubov, N. and Mitropolsky, I., Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.

    Google Scholar 

  7. Kevorkian, J. and Cole, J. D., Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.

    Google Scholar 

  8. Barkham, P. G. D. and Soudack, A. C., 'An extension to the method of Krylov and Bogolioubov', International Journal of Control 10, 1969, 377-392.

    Google Scholar 

  9. Barkham, P. G. D. and Soudack, A. C., 'Approximate solutions of non-linear non-autonomous second-order differential equations', International Journal of Control 11, 1970, 101-114.

    Google Scholar 

  10. Soudack, A. C. and Barkham, P. G. D., 'Further results on approximate solutions of non-linear, nonautonomous second-order differential equations', International Journal of Control 12, 1970, 763-767.

    Google Scholar 

  11. Soudack, A. C. and Barkham, P. G. D., 'On the transient solution of the unforced Duffing equation with large damping', International Journal of Control 13, 1971, 767-769.

    Google Scholar 

  12. Yuste, S. B. and Bejarano, J. D., 'Extension and improvement to the Krylov-Bogolioubov methods using elliptic functions', International Journal of Control 49, 1989, 1127-1141.

    Google Scholar 

  13. Bejarano, J. D., Sanchez, A. M., and Rodriguez, C. M., 'Osciladores alineales excitados no linealmente', Annales de Fisica 78A, 1982, 159-164.

  14. Yuste, S. B. and Bejarano, J. D., 'Amplitude decay of damped non-linear oscillators studied with Jacobian elliptic functions', Journal of Sound and Vibration 114, 1987, 33-44.

    Google Scholar 

  15. Rand, R. H., 'Using computer algebra to handle elliptic functions in the method of averaging', in Symbolic Computations and Their Impact on Dynamics, American Society Mechanical Engineers PVP, Vol. 205, ASME, New York, 1990, pp. 311-326.

    Google Scholar 

  16. Yuste, S. B. and Bejarano, J. D., 'Construction of approximate analytical solutions to a new class of nonlinear oscillator equations', Journal of Sound and Vibration 110, 1986, 347-350.

    Google Scholar 

  17. Garcia-Margallo, J. and Bejarano, J. D., 'Generalized Fourier series and limit cycles of generalized van der Pol oscillators', International Journal of Control 49, 1988, 1127-1141.

    Google Scholar 

  18. Coppola, V. T. and Rand, R. H. 'Averaging using elliptic functions: Approximation of limit cycles', Acta Mechanica 81, 1990, 125-142.

    Google Scholar 

  19. Coppola, V. T. and Rand, R. H., 'Macsyma program to implement averaging using elliptic functions', in Computer Aided Proofs in Analysis, K. R. Meyer and D. S. Schmidt (eds.), Springer-Verlag, Berlin, 1991, pp. 71-89.

    Google Scholar 

  20. Chen, S. H. and Cheung, Y. K., 'An elliptic Lindstedt-Poincaré method for certain strongly non-linear oscillators', Nonlinear Dynamics 12, 1997, 199-213.

    Google Scholar 

  21. Belhaq, M., 'New analytical technique for predicting homoclinic bifurcations in autonomous dynamical systems', Mechanics Research Communications 23(4), 1998, 381-386.

    Google Scholar 

  22. Belhaq, M., Lakrad, F., and Fahsi, A., 'Predicting homoclinic bifurcations in planar autonomous systems', Nonlinear Dynamics 18(4), 1999, 303-310.

    Google Scholar 

  23. Belhaq, M. and Fahsi, A., 'Homoclinic bifurcations in self-excited oscillators', Mechanics Research Communications 23(4), 1996, 381-386.

    Google Scholar 

  24. Xu, Z., Chen, H. S. Y., and Chung, K. W., 'Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method', Nonlinear Dynamics 11, 1996, 213-233.

    Google Scholar 

  25. Chow, N. and Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.

    Google Scholar 

  26. Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995.

    Google Scholar 

  27. Fiedler, B. and Scheurle, J., 'Discretization of homoclinic orbits and invisible chaos', in Memoirs of American Mathematical Society, Vol. 570, American Mathematical Society, Providence, RI, 1996.

    Google Scholar 

  28. Palmer, K. J., 1984, 'Exponential dichotomies and transversal homoclinic points', Journal of Differential Equations 55, 1984, 225-256.

    Google Scholar 

  29. Vanderbauwhede, A., Local Bifurcation and Symmetry, Pitman, Boston, MA, 1982.

    Google Scholar 

  30. Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983.

    Google Scholar 

  31. Dumortier, F., Roussarie, R., and Sotomayor, J., 'Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3', Ergodic Theory and Dynamical Systems 7, 1987, 375-413.

    Google Scholar 

  32. Dangelmayr, G., Fiedler, B., Kirchgässner, K., and Mielke, A., Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Pitman Research Notes on Mathematics, Vol. 352, Addison Wesley, New York, 1996.

    Google Scholar 

  33. Rodriguez-Luis, A. J., Freire, E., and Ponce, E., On a Codimension 3 Bifurcation Arising in an Autonomous Electronic Circuit, International Series of Numerical Mathematics, Vol. 97, Birkhäuser Verlag, Basel, 1991.

    Google Scholar 

  34. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, Berlin, 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belhaq, M., Fiedler, B. & Lakrad, F. Homoclinic Connections in Strongly Self-Excited Nonlinear Oscillators: The Melnikov Function and the Elliptic Lindstedt–Poincaré Method. Nonlinear Dynamics 23, 67–86 (2000). https://doi.org/10.1023/A:1008316010341

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008316010341

Navigation