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Hopf bifurcation at infinity with discontinuous nonlinearities

Published online by Cambridge University Press:  17 February 2009

Xiangjian He
Xiangjian He
Affiliation:
School of Information Sc. and Tech., Flinders University of S. A., Adelaide S.A., Australia.
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Abstract

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In this paper, we consider the existence of a family of periodic solutions of large amplitude when a pair of eigenvalues of the linear part of a first-order system of ordinary differential equations crosses the imaginary axis. We refer to this problem as a Hopf bifurcation problem at infinity. In our work, the nonlinearities may be discontinuous at the origin, and the proof of existence of periodic solutions is arrived at through the corresponding system of integral equations. The applicability of the result is demonstrated by the study of the dynamics of a train truck wheelset system.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 2 , October 1991, pp. 133 - 148
Copyright
Copyright © Australian Mathematical Society 1991

References

[1] Hopf, E., “Abzweigung einer periodischen Lösung von einer stationären Lösung eines differential systems”, Ber. Sächs. Akad. Wiss. Leipzig. Math. Phys. Kl. 95 (1942) 3–22.Google Scholar
[2] Friedrichs, K. O., Advanced ordinary differential equations (Gordon and Breach, New York, 1965).Google Scholar
[3] Chafee, N., “The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system”, J. Differential Equations 4 (1968) 661–679.CrossRefGoogle Scholar
[4] Brushinskaya, N. N., “Qualitative integration of a system of n differential equations in a region containing a singular point and a limit cycle”, Dokl. Akad. Nauk SSSR 139 (1961) 9–12.Google Scholar
[5] Mees, A I. and Chua, L. O., “The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems”, IEEE Transactions on Circuits and Systems, CAS-26 (1979) 235–254.CrossRefGoogle Scholar
[6] Allwright, D. J., “Harmonic balance and the Hopf bifurcation”, Math. Proc. Camb. Phil. Soc. 82 (1977) 453–467.CrossRefGoogle Scholar
[7] Henry, D., “Geometric theory of semilinear parabolic equations”, Lecture Notes in Math. No. 840 (1981).CrossRefGoogle Scholar
[8] Iooss, G., “Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d'evolution du type Navier-Stokes”, Arch. Rational Mech. Anal. 47 (1972) 301–329.CrossRefGoogle Scholar
[9] Fife, P. C., “Branching phenomena in fluid dynamics and chemical reaction-diffusion theory”, Proc. Symp. Eigenvalues of Nonlinear Problems (1974) 23–83.Google Scholar
[10] Joseph, D. D. and Sattinger, D. H., “Bifurcating time periodic solutions and their stability”, Arch Rational Mech. Anal. 45 (1972) 79–109.CrossRefGoogle Scholar
[11] Sattinger, D. H., “Bifurcation of periodic solutions of the Navier-Stokes equations”, Arch. Rational Mech. Anal. 41 (1971) 66–80.CrossRefGoogle Scholar
[12] Kozjakin, V. S. and Krasnosel'skii, M. A., “The method of parameter functionalization in the Hopf bifurcation problem”, Nonlinear Analysis, Theory, Methods and Applications 11 (1987) 149–161.Google Scholar
[13] Glover, J. N., “Hopf bifurcations at infinity”, Nonlinear Analysis, Theory, Methods and Applications, 11 (1989) 1393–1398.Google Scholar
[14] D'Souza, A. F. and Caravavatna, P., “Analysis of nonlinear hunting vibrations of rail vehicle trucks”, ASME, Journal of Mechanical Design 102 (1980) 77–85.CrossRefGoogle Scholar
[15] Blader, F. B. and Kurtz, E. F., “Dynamic stability of cars in long freight trains”, ASME, Journal of Engineering for Industry 96 (1974) 1159–1167.CrossRefGoogle Scholar
[16] Hull, R. and Cooperrider, N. K., “Influence of nonlinear wheel/rail contact geometry on stability of rail vehicles”, ASME, Journal of Engineering for Industry 99 (1977) 172–185.CrossRefGoogle Scholar
[17] Glover, J. N., “Bifurcations and chaos in Hadden's model of a train wheelset”, M.Sc. Thesis, Dept. of Mathematics, University of Western Australia, 1989.Google Scholar