Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T15:31:48.445Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (zhuchangrong126@126.com)
Get access Rights & Permissions [Opens in a new window]

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 

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.)