Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-20T22:53:34.779Z Has data issue: false hasContentIssue false
  • English
  • Français

A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems

Published online by Cambridge University Press:  14 November 2011

James R. Ward Jr.
James R. Ward Jr.
Affiliation:
Department of Mathematics, The University of Alabama at Birmingham, Birmingham, Alabama 35294, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

A general continuation theorem for isolated sets in infinite-dimensional dynamical systems is proved for a class of semiflows. This result is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction—diffusion equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 4 , 1996 , pp. 725 - 738
Copyright
Copyright © Royal Society of Edinburgh 1996

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

1Brown, R. F.. A Topological Introduction to Nonlinear Analysis (Boston: Birkhäuser, 1993).CrossRefGoogle Scholar
2Conley, C. C.. Isolated Invariant Sets and the Morse Index, CBMS 38 (Providence, RI: American Mathematical Society, 1978).CrossRefGoogle Scholar
3Hale, J. K.. Asymptotic Behavior of Dissipative Systems (Providence, RI: American Mathematical Society, 1988).Google Scholar
4Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
5Krasnosel'skii, M. A.. Topological Methods in the Theory of Nonlinear Integral Equations (New York: MacMillan, a Pergamon Press book, 1964).Google Scholar
6Martelli, M.. Continuation principles and boundary value problems. In Topological Methods for Ordinary Differential Equations, eds Furi, M. and Zecca, P., 1537 (Berlin: Springer, 1993).CrossRefGoogle Scholar
7Peitgen, H. O. and Schmitt, K.. Global analysis of two-parameter elliptic eigenvalue problems. Trans. Amer. Math. Soc. 283 (1984), 57–95.CrossRefGoogle Scholar
8Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Fund. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
9Rabinowitz, P. H.. On bifurcation from infinity. J. Differential Equations 14 (1973), 462–75.CrossRefGoogle Scholar
10Rybakowski, K. P.. On the homotopy index for infinite dimensional semiflows. Trans. Amer. Math. Soc. 269(1982), 351–82.CrossRefGoogle Scholar
11Rybakowski, K. P.. The Homotopy Index and Partial Differential Equations (Berlin: Springer, 1987).CrossRefGoogle Scholar
12Schmitt, K. and Wang, Z. Q.. On bifurcation from infinity for potential operators. Differential Integral Equations 4 (1991), 933–43.CrossRefGoogle Scholar
13Toland, J. F.. Asymptotic linearity and nonlinear eigenvalue problems. Quat. J. Math. Oxford Ser. (2) 24 (1973), 241–50.CrossRefGoogle Scholar
14Toland, J. F.. Bifurcation and asymptotic bifurcation for non-compact non-symmetric gradient operators. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 137–47.CrossRefGoogle Scholar
15Ward, J. R. Jr., Bifurcating continua in infinite dimensional dynamical systems and applications to differential equations, J. Diff. Equations (to appear).Google Scholar
16Whyburn, G. T.. Topological Analysis (Princeton, NJ: Princeton University Press, 1958).Google Scholar
17Zeidler, E.. Nonlinear Functional Analysis and its Applications, vol. I: Fixed-Point Theorems (Berlin: Springer, 1985).CrossRefGoogle Scholar