Abstract
In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation \(\ddot X + 9X + \mu X^2 + {\lambda }X^3 = \varepsilon (\dot X - \dot X^3 )\) will be studied in detail, and it will beshown that at least five limit cycles can occur.
Similar content being viewed by others
References
Waluya, S. B. and Van Horssen, W. T., 'Asymptotic approximations of first integrals for a nonlinear oscillator', Nonlinear Analysis 51(8), 2002, 1327–1346.
Waluya, S. B. and Van Horssen, W. T., 'On approximations of first integrals for a system of weakly nonlinear, coupled harmonic oscillators', Nonlinear Dynamics 30, 2002, 243–266.
Van Horssen, W. T., 'On integrating factors for ordinary differential equations', Nieuw Archief voor Wiskunde 15, 1997, 15–26.
Van Horssen, W. T., 'A perturbation method based on integrating factors', SIAM journal on Applied Mathematics 59(4), 1999, 1427–1443.
Van Horssen, W. T., 'A perturbation method based on integrating vectors and multiple scales', SIAM Journal on Applied Mathematics 59(4), 1999, 1444–1467.
Van Horssen, W. T., 'On integrating vectors and multiple scales for singularly perturbed ordinary differential equations', Nonlinear Analysis TMA 46, 2001, 19–43.
Doelman, A. and Verhulst, F., 'Bifurcations of strongly non-linear self-excited oscillations', Mathematical Methods in the Applied Sciences 17, 1994, 189–207.
Yuste, S. B. and Bejarano, J. D., 'Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions', International Journal of Control 49(4), 1989, 1127–1141.
Coppola, V. T. and Rand, R. H., 'Averaging using elliptic functions: Approximation of limit cycles', Acta Mechanica 81, 1990, 125–142.
Roy, R. V., 'Averaging methods for strongly non-linear oscillators with periodic excitations', International Journal of Non-Linear Mechanics 29(5), 1994, 737–753.
Chen, S. H. and Cheung, Y. K., 'An elliptic Lindstedt–Poincaré method for certain strongly non-linear oscillators', Nonlinear Dynamics 12, 1997, 199–213.
Chen, S. H. and Cheung, Y. K., 'An elliptic perturbation method for certain strongly non-linear oscillators', Journal of Sound and Vibration 192(2), 1996, 453–464.
Iliev, I. D. and Perko, L. M., 'Higher order bifurcations of limit cycles', Journal of Differential Equations 154, 1999, 339–363.
Blows, T. R. and Perko, L.M., 'Bifurcation of limit cycles from center and sparatrix cycles of planar analytic systems', SIAM Review 36(3), 1994, 341–376.
Margallo, J. G. and Bejarano, J. D., 'Stability of limit cycles and bifurcations of generalized van der Pol oscillator: \({\ddot X}\)+AX–2BX 3+\(\varepsilon \)(z 3+z 2 X 2+z 1 X 4) \({\dot X}\) = 0', International Journal of Non-Linear Mechanics 25(6), 1990, 663–675.
Margallo, J. G. and Bejarano, J. D., 'The limit cycles of the generalized Rayleigh–Liénard oscillator', Journal of Sound and Vibration 156(2), 1992, 283–301.
Lynch, S., 'Small amplitude limit cycles of the generalized mixed Rayleigh–Liénard oscillator', Journal of Sound and Vibration 178(5), 1994, 615–620.
Van der Beek, C. G. A., 'Normal form and periodic solutions in the theory of non-linear oscillations. Existence and asymptotic theory', International Journal of Non-Linear Mechanics 24(4), 1989, 263–279.
Van der Beek, C. G. A., 'Analysis of a system of two weakly nonlinear coupled harmonic oscillators arising from the field of wind-induced vibrations', International Journal of Non-Linear Mechanics 27(4), 1992, 679-704.
Brothers, J. D. and Haberman, R., 'Accurate phase after passage through subharmonic resonance', SIAM Journal of Applied Mathematics 59(1), 1998, 347–364.
Bosley, D. L., 'An improved matching procedure for transient resonance layers in weakly nonlinear oscillatory systems', SIAM Journal of Applied Mathematics 56(2), 1996, 420–445.
Lawden, D. F., Elliptic Functions and Applications, Springer-Verlag, New York, 1989.
Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn., Springer, Berlin, 1971.
Van Horssen, W. T. and Kooij, R. E., 'Bifurcation of limit cycles in a particular class of quadratic systems with two centers', Journal of Differential Equations 114(2), 1994, 538–569.
Van Horssen, W. T. and Reyn, J. W., 'Bifurcation of limit cycles in a particular class of quadratic systems', Differential and Integral Equations 8(4), 1995, 907–920.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Waluya, S.B., van Horssen, W.T. On Approximations of First Integrals for Strongly Nonlinear Oscillators. Nonlinear Dynamics 32, 109–141 (2003). https://doi.org/10.1023/A:1024470410240
Issue Date:
DOI: https://doi.org/10.1023/A:1024470410240