From homoclinics to quasi-periodic solutions for ordinary differential equations

Changrong Zhu

doi:10.1017/S0308210515000189

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

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Changrong Zhu*: Affiliation:
School of Mathematics and Physics, Chongqing University, Chongqing 400044, People’s Republic of China (zhuchangrong126@126.com)

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Abstract

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Abstract

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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.