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A shadowing approach to passage through resonance

Published online by Cambridge University Press:  14 November 2011

James A. Murdoch
James A. Murdoch
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A.
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Synopsis

Leading order approximations are given by a patching method for passage through resonance in the case when the resonance zone contains saddle points. The approximations are uniformly valid regardless of the length of time required to pass through the resonance. Accuracy for extended time periods is obtained by asking not for approximate solutions with specified initial values, but for approximate solutions which are “shadowed” by exact solutions in the resonance zone.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 1-2 , 1990 , pp. 1 - 22
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Chow, Shui-Nee, Lin, Xiao-Biao and Palmer, Kenneth J.. A shadowing lemma with applications to semilinear parabolic equations (preprint).Google Scholar
2Kevorkian, J.. Perturbation techniques for oscillatory systems with slowly varying coefficients. SIAM Rev. 29, 1987, 391–460.CrossRefGoogle Scholar
3Lochak, P. and Meunier, C.. Multiphase Averaging for Classical Systems (New York: Springer, 1988).CrossRefGoogle Scholar
4Murdock, James. Qualitative theory of nonlinear resonance by averaging and dynamical systems methods. In Dynamics Reported, vol. 1, eds Kirchgraber, V. and Walther, H. O., pp. 91–172 (New York: Wiley, 1988).CrossRefGoogle Scholar
5Murdock, James. On the length of validity of averaging and the uniform approximation of elbow orbits, with an application to delayed passage through resonance. Z. Angew. Math. Phys. 39 (1988), 586–596.Google Scholar
6Murdock, James and Robinson, Clark. Qualitative dynamics from asymptotic expansions: local theory. J. Differential Equations 36 (1980), 425–441.Google Scholar
7Sanders, J. A.. On the passage through resonance. SIAM J. Math. Anal. 10 (1979), 1220–1243.Google Scholar
8Sanders, J. A. and Verhulst, F.. Averaging Methods in Nonlinear Dynamical Systems. (New York: Springer, 1985).Google Scholar