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degenerate homoclinic solution

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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1 results

From homoclinics to quasi-periodic solutions for ordinary differential equations
Part of
- Parabolic equations and systems
Changrong Zhu
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 145 / Issue 5 / October 2015

Published online by Cambridge University Press:

01 September 2015, pp. 1091-1114

Print publication:

October 2015
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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.