×

A hyperbolic Lindstedt-Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators. (English) Zbl 1269.70031

Summary: A hyperbolic Lindstedt-Poincaré method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Liénard oscillator is studied in detail, and the present method’s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.

MSC:

70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chen S.H., Chen Y.Y., Sze K.Y.: A hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators. J. Sound Vib. 322, 381–392 (2009)
[2] Wang Z.H., Hu H.Y.: A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator. Acta Mech. Sin. 24(4), 449–454 (2008) · Zbl 1271.70053
[3] Gan C.B., He S.M.: Studies on structural safety in stochastically excited Duffing oscillator with double potential wells. Acta Mech. Sin. 23(5), 577–583 (2007) · Zbl 1202.70120
[4] 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)
[5] Chan H.S.Y., Chung K.W., Xu Z.: Stability and bifurcations of limit cycles by the perturbation-incremental method. J. Sound Vib. 206, 589–604 (1997) · Zbl 1235.34093
[6] Chen S.H., Chan J.K.W., Leung A.Y.T.: A perturbation method for the calculation of semi-stable limit cycles of strongly nonlinear oscillators. Commun. Numer. Methods Eng. 16, 301–313 (2000) · Zbl 0964.65145
[7] Zhang Y.M., Lu Q.S.: Homoclinic bifurcation of strongly nonlinear oscillators by frequency-incremental method. Commun. Nonlinear Sci. Numer. Simul. 8(1), 1–7 (2000) · Zbl 1027.34053
[8] Zhang Q., Wang W., Li W.: Heteroclinic bifurcations of strongly nonlinear oscillator. Chin. Phys. Lett. 25(5), 1905–1907 (2008)
[9] Belhaq M.: Predicting homoclinic bifurcations in planar autonomous systems. Nonlinear Dyn. 18, 303–310 (1999) · Zbl 0943.34026
[10] Belhaq M., Lakrad F.: Prediction of homoclinic bifurcation: the elliptic averaging method. Chaos Solitons Fract. 11, 2251–2258 (2000) · Zbl 0953.34026
[11] Belhaq M., Fiedler B., Lakrad F.: Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt–Poincaré method. Nonlinear Dyn. 23, 67–86 (2000) · Zbl 0967.70019
[12] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[13] Nayfeh A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) · Zbl 0449.34001
[14] Merkin J.H., Needham D.J.: On infinite period bifurcations with an application to roll waves. Acta Mech. 60, 1–16 (1986) · Zbl 0588.76024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.