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Bifurcation from a homoclinic orbit in parabolic differential equations

Published online by Cambridge University Press:  14 November 2011

C. Miguel Blázquez
C. Miguel Blázquez
Affiliation:
Brown University, Division of Applied Mathematics, Providence, R. I. 02912, U.S.A.
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Synopsis

This paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 265 - 274
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Andronov, A., Leontovich, E. et al. Theory of Bifurcations in the Plane (New York: Wiley, 1973).Google Scholar
2Blázquez, C. M.. Transverse homoclinic orbits in periodically perturbed parabolic equations. LCDS Report 85–22 J. Nonlinear Anal. 6 (1986).Google Scholar
3Chow, S. M., Hale, J. K. and Mallet-Paret, J.. An example of bifurcation to homoclinic orbits. J. Differential Equations 37 (1980), 351–373.CrossRefGoogle Scholar
4Conley, C. C.. On travelling wave solutions of nonlinear diffusion equations. Lectures Notes in Physics 38 (Berlin: Springer, 1975).Google Scholar
5Friedman, A.. Partial Differential Equations (New York: Holt, Reinhart and Winston, 1969).Google Scholar
6Henry, D.. Geometric Theory of Semilinear Parabolic Equations. Lecture Note in Mathematics 840 (Berlin: Springer, 1981).Google Scholar
7Holmes, P. and Marsden, J.. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Rational Mech. Anal. 76 (1981), 135–165.Google Scholar
8Lin, X. B.. Exponential dichotomies and homoclinic orbits in functional differential equations LCDS Report 84–17 (Providence, R. I.: Brown University, 1984).Google Scholar
9Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sc. 44 (Berlin: Springer, 1983).CrossRefGoogle Scholar
10Sil'nikov, L. P.. Some cases of generation of periodic motions from singular trajectories. Mat. Sb. (N.S.) 61 (103) (1963), 443–466.Google Scholar
11Sil'nikov, L. P.. A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR 160 (1965), 558–561; Soviet Math. Dokl. 6 (1965).Google Scholar
12Sil'nikov, L. P.. On a Poincaré-Birkhoff problem. Mat. Sb. N.S. 74 (116), (1967), 378–397; Math. USSR-Sb. 3 (1967), 353–371.Google Scholar
13Smoller, J.. Shock Waves and Reaction-Diffusion Equations (New York: Springer, 1983).CrossRefGoogle Scholar