Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-27T03:22:26.836Z Has data issue: false hasContentIssue false

Bifurcation from homoclinic to periodic solutions in singularly perturbed differential inclusions

Published online by Cambridge University Press:  14 November 2011

Michal Fečkan
Affiliation:
Department of Mathematical Analysis, Comenius University, Mlynská dolina, 842 15 Bratislava, Slovakia

Abstract

The existence of periodic solutions is studied for certain singularly perturbed differential inclusions. Applications are given to dry friction problems.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Andronow, A. A., Witt, A. A. and Chaikin, S. E.. Theorie der Schwingungen I (Berlin: Akademie, 1965).Google Scholar
2Battelli, F. and Lazzari, C.. Bounded solutions to singularly perturbed systems of O.D.E. J. Differential Equations 100 (1992), 4981.CrossRefGoogle Scholar
3Deimling, K.. Multivalued Differential Equations (Berlin: W. De Gruyter, 1992).CrossRefGoogle Scholar
4Deimling, K.. Multivalued differential equations and dry friction problems. In Proceedings of the Conference on Differential and Delay Equations, Ames, Iowa 1991, eds Fink, A. M., Miller, R. K. and Kliemann, W., 99106 (Singapore: World Scientific, 1992).Google Scholar
5Deimling, K. and Szilágyi, P.. Periodic solutions of dry friction problems. Z. Angew. Math. Phys. 45 (1994), 5360.CrossRefGoogle Scholar
6Hartog, J. P. Den. Mechanische Schwingungen, 2nd edn (Berlin: Springer, 1952).CrossRefGoogle Scholar
7Fečkan, M.. Melnikov functions for singularly perturbed ordinary differential equations. Nonlinear Anal. 19 (1992), 393401.CrossRefGoogle Scholar
8Fečkan, M.. Bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. J. Differential Equations 130 (1996), 415–50.CrossRefGoogle Scholar
9Fečkan, M. and Gruendler, J.. Bifurcation from homoclinic to periodic solutions in ordinary differential equations with singular perturbations (submitted).Google Scholar
10Gruendler, J.. Chaotic regions for singularly perturbed systems of ordinary differential equations (Preprint).Google Scholar
11Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer, 1983).CrossRefGoogle Scholar
12Kauderer, H.. Nichtlineare Mechanik (Berlin: Springer, 1958).CrossRefGoogle Scholar
13Palmer, K. J.. Exponential dichotomies and transversed homoclinic points. J. Differential Equations 55(1984), 225–56.CrossRefGoogle Scholar
14Pruszko, T.. Topological degree methods in multi-valued boundary value problems. Nonlinear Anal. 5(1981), 959–73.CrossRefGoogle Scholar
15Pruszko, T.. Some applications of the topological degree theory to multi-valued boundary value problems. Dissertationes Math. 229 (1984), 148.Google Scholar
16Yosida, K.. Functional Analysis (Berlin: Springer, 1965).Google Scholar