Bifurcations from homoclinic orbits to non-hyperbolic equilibria in reversible lattice differential equations
Bifurcations from homoclinic orbits to non-hyperbolic equilibria in reversible lattice differential equations
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DOI：
10.1088/0951-7715/21/4/005
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发表时间：
2008-04
期刊：
影响因子：
1.7
通讯作者：
M. Georgi
中科院分区：
文献类型：
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作者：
M. Georgi
从一个格点微分方程（LDE）出发，研究行波解，得到一个向前向后延迟方程。存在于LDE中的结构通常由行波方程继承。在本文中，我们感兴趣的情况下，行波方程是可逆的，并拥有一个对称的同宿解。此外，我们考虑渐近稳态恰好有两个纯虚特征值±iω，ω <$0的情况。因此，一族小的周期解存在附近的稳定状态。这是本文的目的是分析这种分歧下的一般假设，并利用基本的可逆性方程。作为主要结果之一，我们发现所有的同宿轨道的中心流形，这接近一个周期轨道的向前和向后的时间。
Starting with a lattice differential equation (LDE), the study of travelling wave solutions leads to a forward–backward delay equation. Structures which are present in the LDE typically are inherited by the travelling wave equation. In this paper we are interested in a situation where the travelling wave equation is reversible and possesses a symmetric homoclinic solution. Moreover, we consider the case that the asymptotic steady state possesses exactly two purely imaginary eigenvalues ±iω, ω ≠ 0. As a consequence, a family of small periodic solutions exist near the steady state. It is the aim of this paper to analyse this bifurcation under generic assumptions and to exploit the underlying reversibility of the equation. As one of the main results we find all homoclinic orbits to the centre manifold, which approach a periodic orbit in forward and backward time.