Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-25T01:21:37.159Z Has data issue: false hasContentIssue false
  • English
  • Français

Hopf Bifurcation for Implicit Neutral Functional Differential Equations

Published online by Cambridge University Press:  20 November 2018

Tomasz Kaczynski  and
Huaxing Xia
Tomasz Kaczynski
Affiliation:
Département de mathématiques et d'informatique Université de Sherbrooke Sherbrooke, Québec J1K 2R1
Huaxing Xia
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

Keywords

34K4034A0934C2358E09
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 286 - 295
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bielawski, R. and Gorniewicz, L., A fixed point index approach to some differential equations. In: Topological Fixed Point Theory and Applications, (ed. Boju Jiang), Lecture Notes in Mathematics (1411), Springer 1989,9–14.Google Scholar
2. Bielawski, R. and Gorniewicz, L., Some applications of the Leray-Schauder alternative to differential equations, (ed. S. P. Singh), Reidel, 1986,187–194.Google Scholar
3. Bredon, G. E., Introduction to Compact Transformation Groups, Academic Press, NY-London, 1972.Google Scholar
4. Chow, S. N. and Hale, J. K., Method of Bifurcation Theory, Springer-Verlang, NY, 1982.Google Scholar
5. Dugundji, J. and Granas, A., Fixed Point Theory I, PWN, Warszawa 1981.Google Scholar
6. Dylawerski, G., Geba, K., Jodel, J. and Marzantowicz, W., An Sl -equivariant degree and the Fuller index, Ann. Pol. Math. (52) 3(1991), 243–280.Google Scholar
7. Erbe, L. H., Geba, K., Krawcewicz, W. and Wu, J., Sl -degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations (98) 2(1992), 277–298.Google Scholar
8. 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(t1y',yn), Rocky Mountain J. Math. (20) 4(1990), 899–907.Google Scholar
9. Frigon, M. and Kaczynski, T., Boundary value problems for systems of implicit differential equations, J. Math. Anal, and Appl, to appear.Google Scholar
10. Geba, K. and Marzantowicz, W., Global bifurcation of periodic solutions, Topological Methods in Nonlinear Analysis, to appear.Google Scholar
11. Haddad, G., Topological properties of the sets of solutions for functional-differential inclusions, Nonlinear Analysis TMA 5(1981), 1349–1366.Google Scholar
12. Haddad, G. and Lasry, J. M., Periodic solutions of functional differential inclusions and fixed points of G-selectionable correspondences J. Math. Anal, and Appl. 96(1983), 295–312.Google Scholar
13. Hale, J. K., Nonlinear oscillations in equations with delays. In: Nonlinear Oscillations in Biology, Lectures in Applied Math. 17, Amer. Math. Soc, Providence, R.I., (1978), 157-189,.Google Scholar
14. Hassard, B., Kazarinoff, N. and Wan, Y., Theory and Applications of Hopf Bifurcation, Cambridge, 1981.Google Scholar
15. Ize, J., Bifurcation Theory for Fredholm Operators, Mem. Amer. Math. Soc. (174), (1976).Google Scholar
16. Kaczynski, T., Implicit differential equations which are not solvable for the highest derivative. In: Delay Differential Equations and Dynamical Systems, (ed. S. Busenberg and M. Martelli), Lecture Notes in Math. (1475), Springer-Verlag, 1991,218–224.Google Scholar
17. Kaczynski, T. and Krawcewicz, W., Bifurcation problem for a certain class of implicit differential equations, Canad. Math. Bull. 36(1993), 183–189.Google Scholar
18. Kaczynski, T. and Wu, J., A topological transversality theorem for multivalued maps in locally convex spaces with applications to neutral equations, Can. J. Math. 44(1992), 1003–1013.Google Scholar
19. Kisielewicz, M., Existence theorem for generalized functional differential equations of neutral type, J. Math. Anal, and Appl. 78(1980), 173–182.Google Scholar
20. Krawcewicz, W., Xia, H. and Wu, J., Sl -equivariant degree and global Hopf bifurcation for condensing fields with applications to neutral functional differential equations, Canad. Appl. Math. Quart. 1(1993), 1–54.Google Scholar
21. Marsden, J. and F McCracken, M., The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976.Google Scholar
22. Nussbaum, R. D., A Hopf global bifurcation theoremfor retartedfunctional differential equations, Trans. Amer. Math. Soc. 238(1976), 139–164.Google Scholar
23. F, J. C. DeOliverira, Hopf bifurcation for functional differential equations, Nonlinear Analysis, TMA 4(1980), 217–229.Google Scholar
24. Papageorgiou, N. S., On the theory of Junctional differential inclusions of neutral type in Banach space, Func. Ekvac. 31(1988), 103–120.Google Scholar
25. Pruszko, T., Topological degree methods in multi-valued boundary problems, Nonlinear Analysis, TMA (9) 5(1981), 959–973.Google Scholar
26. Rustichini, A., Hopf bifurcation for functional differential equations of mixed type, J. Dynamics and Differential Equations 1(1989), 145.Google Scholar
27. Sadovskij, B. N., Limit-compact and condensing operators, UspehiMat. Nauk 27(1971), 81–146.Google Scholar
28. Staffans, O. J., Hopf bifurcation for functional differential equations with infinite delay, J. Differential Equations 70(1987), 114–151.Google Scholar
29. Stech, H. W, Hopf bifurcation for Junctional differential equations, SIAM J. Math. Anal. 16(1985), 1134–1151.Google Scholar
30. Xia, H., S -equivariant bijurcation theory for multivalued mappings and its applications to neutral functional differential inclusions, preprint.Google Scholar