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A Local Hopf Bifurcation Theorem for a Certain Class of Implicit Differential Equations

Published online by Cambridge University Press:  20 November 2018

Tomasz Kaczynski  and
Wieslaw Krawcewicz
Tomasz Kaczynski
Affiliation:
Département de mathématiques et d'informatique Université de Sherbrooke Sherbrooke, Québec J1K 2R1
Wieslaw Krawcewicz
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1
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Abstract

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The local Hopf Bifurcation theorem is extended to implicit differential equations in Rn, of the form ẋ = f(x,ẋ, α), which are not solvable for the variable ẋ. The proof uses the S1 -degree of convex-valued mappings. An example of an implicit differential equation in R3 to which the presented theorem applies is provided.

Keywords

34A0934C2347H1558E09
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 2 , 01 June 1993 , pp. 183 - 189
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bielawski, R. and Gorniewicz, L., A fixed point index approach to some differential equations, Proceedings of the conference Topological Fixed Point Theory and Applications, (ed. Boju Jiang), Lecture Notes in Math. (1411), Springer, 1989,9–14.Google Scholar
2. Chow, S. N. and Hale, J., Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.Google Scholar
3. Dylawerski, G., Geba, K., Jodel, J. and Marzantowicz, W., An S1 -equivariant degree and the Fuller index, Ann. Pol. Math., 52(1991), 243–279.Google Scholar
4. Erbe, L. H., Kaczynski, T. and Krawcewicz, W., Solvability of two point boundary value problems for systems of nonlinear differential equations of the form y″ = g(t,y,y′ ,y″), Rocky Mountain J.of Math. (4)20(1990), 899–907.Google Scholar
5. Erbe, L. H., Geba, K., Krawcewicz, W. and Wu, J., Sl -degree and global Hopf bifurcation theory of functional differential equations,]. Diff. Eq. (2) 98(1992), 277–298.Google Scholar
6. Geba, K. and Marzantowicz, W., Global bifurcation of periodic solutions, Topological Methods in Nonlinear Analysis, to appear.Google Scholar
7. Hopf, E., Abzweigung einer periodischen Lbsung von einer stationàren Losung eines Differential systems, Ber. Math. Phys., Sachsische Akad. der Wissenschaften, Leipzig, 95(1943), 3–22.Google Scholar
8. Kaczynski, T., On differential inclusions of second order, Proceedings of the Equadiff Conference (ed. C. M. Dafermos, G. Ladas and G. Papanicolau), Marcel Dekker, 1989,343–352.Google Scholar
9. Kaczynski, T., Implicit differential equations which are not solvable for the highest derivative, in Delay Differential Equation and Dynamical Systems, Proceedings, Claremont 1990 (eds. S. Busenberg and M. Martelli), Lecture Notes in Math. (1475), Springer-Verlag 1991, 218–224.Google Scholar
10. Kaczynski, T. and Wu, J., A topological transversality theorem for multi-valued maps in locally convex spaces with applications to neutral equations, Canad. J. of Math. (5) 44(1992), 1003–1013.Google Scholar
11. Krawcewicz, W., Spanily, T. and Wu, J., Hopfbifurcation for parametrized equivariant coincidenceproblems and parabolic equations with delays, preprint.Google Scholar
12. Marsden, J. and McCracken, M. F., The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976.Google Scholar
13. Pruszko, T., Topological degree methods in multi-valued boundary value problems, J. Nonlinear Analysis- TMA (9) 5(1981), 959–973.Google Scholar
14. Welsh, S. C., Sufficient conditions for periodic solutions to a class of second-order differential equations, Nonlinear Anal-TM A (1) 17(1991), 85–93.Google Scholar
15. Xia, H., S1 -equivariant bifurcation theory for multivalued mappings and its application to neutral functional differential inclusions, preprint.Google Scholar