Abstract
The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.
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References
Nayfeh, A. H., Perturbation Methods, Wiley-Interscience, New York, 1973.
Barkham, P. G. D. and Soudack, A. C., ‘An extension to the method of Krylov and Bogoliubov’, International Journal of Control 10, 1969, 377–392.
Yuste, S. B. and Bejarano, J. D., ‘Extension and improvement to the Krylov-Bogoliubov methods using elliptic function’, International Journal of Control 49, 1989, 1127–1141.
Coppola, V. T. and Rand, R. H., ‘Averaging using elliptic functions: Approximation of limit cycles’, Acta Mechanica 81, 1990, 125–142.
Margallo, J. G. and Bejarano, S. B., ‘Generalized Fourier series for the study of limit cycles’, Journal of Sound and Vibration 125, 1988, 13–21.
Xu, Z. and Zhang, L. Q., ‘Asymptotic method for analysis of non-linear systems with two parameters’, Acta Mathematica Scientia 6, 1986, 453–462.
Xu, Z., ‘Nonlinear time transformation method for strong nonlinear oscillation systems’, Acta Mechanica Sinica 8, 1992, 279–288.
Xu, Z. and Cheung, Y. K., ‘Averaging method using generalized harmonic functions for strongly non-linear oscillators’, Journal of Sound and Vibration 174, 1994, 563–576.
Xu, Z. and Cheung, Y. K., ‘A non-linear scales method for strongly non-linear oscillators’, Nonlinear Dynamics 7, 1995, 285–299.
Kuzmak, G. E., ‘Asymptotic solutions of nonlinear second order differential equations with variable coefficients’, Prikladnaya Matematika Mekhanika 23, 1959, 515–526.
Summers, J. L. and Savage, M. D., ‘Two timescale harmonic balance, I. Application to autonomous one-dimensional nonlinear oscillators’, Philosophical Transactions of the Royal Society London A 340, 1992, 473–501.
Chan, H. S. Y., Chung, K. W., and Xu, Z., ‘A perturbation-iterative method for determining limit cycles of strongly non-linear oscillators’, Journal of Sound and Vibration 183, 1995, 707–717.
Lau, S. L. and Cheung, Y. K., ‘Amplitude incremental variational principle for nonlinear vibration of elastic system’, ASME Journal of Applied Mechanics 48, 1981, 959–964.
Lau, S. L. and Yuen, S. W., ‘Solution diagram of non-linear dynamic systems by the IHB method’, Journal of Sound and Vibration 167, 1993, 303–316.
Chan, H. S. Y., Chung, K. W., and Xu, Z., ‘A perturbation-incremental method for strongly non-linear oscillators’, International Journal of Non-Linear Mechanics 31, 1996, 59–72.
Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
Merkin, J. H. and Needham, D. J., ‘On infinite period bifurcations with an application to roll waves’, Acta Mechanica 60, 1986, 1–16.
Sansone, G. and Conti, R., Nonlinear Differential Equations, Pergamon Press, London, 1964.
Nocilla, S. and Moroni, P., ‘Existence and nonexistence of periodic solutions for the Liénard and related equations’, Nonlinear Dynamics 6, 1994, 331–352.
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Xu, Z., Chan, H.S.Y. & Chung, K.W. Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method. Nonlinear Dyn 11, 213–233 (1996). https://doi.org/10.1007/BF00120718
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DOI: https://doi.org/10.1007/BF00120718