Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria
Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria
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DOI：
10.1016/j.jde.2012.08.008
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发表时间：
2010-09
期刊：
影响因子：
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通讯作者：
D. Blázquez-Sanz;K. Yagasaki
中科院分区：
文献类型：
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作者：
D. Blázquez-Sanz;K. Yagasaki
我们研究了一类可能是哈密顿的或非哈密顿的四维系统的同宿轨到双曲鞍点平衡点的分支。在哈密顿系统中，只有一个参数就足以处理这些类型的分叉，但在一般系统中需要两个参数。利用Gruendler提出的Melnikov-ʼ-Gruendler方法得到同宿轨的鞍节型和干叉型分支结果。在同宿轨位于解析不变流形上的假设下，证明了如果发生这些分支，则同宿轨周围的变分方程在微分Galois理论意义下是可积的。我们用一个作为耦合的实Ginzburg-Landau偏微分方程定常态的例子来说明我们的理论，并用数值方法证明了理论结果。
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikovʼs method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.