×

Prediction of homoclinic bifurcation: the elliptic averaging method. (English) Zbl 0953.34026

Summary: A criterion to predict bifurcation of homoclinic orbits in strongly nonlinear autonomous oscillators is presented. The averaging method combined formally with the Jacobian elliptic functions is applied to determine an approximation to limit cycles near homoclinicity. The authors introduce a criterion for predicting homoclinic orbits, based on the collision between the bifurcating limit cycle and the saddle equilibrium. In particular, they show that this criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic nonlinearities are included.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C29 Averaging method for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley New York
[2] Jordan DW, Smith P. Nonlinear oordinary differential equations. Oxford: Oxford University Press, 1987
[3] Nayfeh, A.H., Perturbation methods, (1973), Wiley New York · Zbl 0375.35005
[4] Nayfeh, A.H., Introduction to perturbation techniques, (1981), Wiley New York · Zbl 0449.34001
[5] Krylov, N.; Bogolioubov, N., Introduction to nonlinear mechanics, (1943), Princeton University Press Princeton, NJ
[6] Bogolioubov, N.; Mitropolsky, I., Asymptotic methods in the theory of nonlinear oscillations, (1961), Gordon and Breach New York
[7] Kevorkian, J.; Cole, J.D., Perturbation methods in apllied mathematics, (1981), Springer New York · Zbl 0456.34001
[8] Barkham, P.G.D.; Soudack, A.C., An extension to the method of Krylov and bogolioubov, Internat J control, 10, 377-392, (1969) · Zbl 0176.46702
[9] Barkham, P.G.D.; Soudack, A.C., Approximate solutions of nonlinear non-autonomous second-order differential equations, Internat J control, 11, 101-114, (1970) · Zbl 0186.15704
[10] Soudack, A.C.; Barkham, P.G.D., Further results on approximate solutions of nonlinear, non-autonomous second-order differential equations, Internat J control, 12, 763-767, (1970) · Zbl 0202.09701
[11] Soudack, A.C.; Barkham, P.G.D., On the transient solution of the unforced Duffing equation with large damping, Internat J control, 13, 767-769, (1971) · Zbl 0217.28301
[12] Yuste, S.B.; Bejarano, J.D., Extension and improvement to the krylov – bogolioubov methods using elliptic functions, Internat J control, 49, 1127-1141, (1989) · Zbl 0691.34029
[13] Bejarano, J.D.; Sanchez, A.M.; Rodriguez, C.M., Osciladores alineales excitados no linealmente, Annales de fisica A, 78, 159-164, (1982)
[14] Yuste, S.B.; Bejarano, J.D., Amplitude decay of damped nonlinear oscillators studied with Jacobian elliptic functions, J sound vibration, 114, 33-44, (1987) · Zbl 1235.70060
[15] Rand, R.H., Using computer algebra to handle elliptic functions in the method of averaging. symbolic computations and their impact on dynamics, American soc mech engrg PVP, 205, (1990)
[16] Yuste, S.B.; Bejarano, J.D., Construction of approximate analytical solutions to a new class of nonlinear oscillator equations, J sound vibration, 110, 347-350, (1986) · Zbl 1235.70146
[17] Garcia-Margallo, J.; Bejarano, J.D., Generalized Fourier series and limit cycles of generalized van der Pol oscillators, Internat J control, 49, 1127-1141, (1988)
[18] Coppola, V.T.; Rand, R.H., Averaging using elliptic functions: approximation of limit cycles, Acta mech, 81, 125-142, (1990) · Zbl 0699.34032
[19] Coppola VT, Rand RH. Macsyma program to implement averaging using elliptic functions. Computer aided proofs in analysis. Germany: Springer, 1991;71-89
[20] Chen, S.H.; Cheung, Y.K., An elliptic lindstedt – poincaré method for certain strongly nonlinear oscillators, Nonlinear dynamics, 12, 199-213, (1997) · Zbl 0881.70015
[21] Chen, S.H.; Cheung, Y.K., An elliptic perturbation method for certain strongly nonlinear oscillators, J sound vibration, 192, 2, 453-464, (1996) · Zbl 1232.70017
[22] Belhaq, M., New analytical technique for predicting homoclinic bifurcations in autonomous dynamical systems, Mech res comm, 23, 4, 381-386, (1998) · Zbl 0934.37013
[23] Belhaq, M.; Fahsi, A.; Lakrad, F., Predicting homoclinic bifurcations in planar autonomous systems, Nonlinear dynamics, 18, 4, 303-310, (1999) · Zbl 0943.34026
[24] Belhaq, M.; Fahsi, A., Homoclinic bifurcations in self-excited oscillators, Mech res comm, 23, 4, 381-386, (1996) · Zbl 0900.70318
[25] Belhaq M, Fiedler B, Lakrad F. Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function on the elliptic Lindstedt-Poincaré method, Nonlinear dynamics, to appear · Zbl 0967.70019
[26] Belhaq M, Houssni M, Freire E, Rodríguez-Luis AJ. Asymptotics of homoclinic bifurcation in a three-dimensional system. Nonlinear Dynamics, to appear · Zbl 0960.70019
[27] Chow, N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer New York · Zbl 0487.47039
[28] Kuznetsov, Y.A., Elements of applied bifurcation theory, (1995), Springer New York · Zbl 0829.58029
[29] Guckenheimer, J.; Holmes, P.J., Nonlinear oscillations, dynamical systems and bifurcation of vector fields, (1983), Springer New York
[30] 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
[31] Byrd, P.; Friedman, M., Handbook of elliptic integrals for engineers and scientists, (1971), Springer Berlin · Zbl 0213.16602
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.