Analytic and Algebraic Conditions for Bifurcations of Homoclinic Orbits II: Reversible Systems

被引:0
作者
Kazuyuki Yagasaki
机构
[1] Kyoto University,Department of Applied Mathematics and Physics, Graduate School of Informatics
来源
Journal of Dynamics and Differential Equations | 2023年 / 35卷
关键词
Homoclinic orbit; Bifurcation; Reversible system; Differential Galois theory; Melnikov method; 34C23; 34C37; 37C29; 34A30;
D O I
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学科分类号
摘要
Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.
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页码:1863 / 1884
页数:21
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