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From homoclinics to quasi-periodic solutions for ordinary differential equations

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  01 September 2015

Changrong Zhu
Changrong Zhu*
Affiliation:
School of Mathematics and Physics, Chongqing University, Chongqing 400044, People’s Republic of China ([email protected])
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Abstract

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

Keywords

degenerate homoclinic solutionquasi-periodic solutionLyapunov–Schmidt reductionFredholm alternativeexponential dichotomy

MSC classification

Primary: 35K65: Degenerate parabolic equations
Secondary: 35K70: Ultraparabolic equations, pseudoparabolic equations, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 5 , October 2015 , pp. 1091 - 1114
Copyright
Copyright © Royal Society of Edinburgh 2015 

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