Subsidiary homoclinic orbits to a saddle focus for reversible systems
Subsidiary homoclinic orbits to a saddle focus for reversible systems
复制标题
可逆系统鞍焦点的辅助同宿轨道
DOI：
--
复制
发表时间：
1994
期刊：
影响因子：
--
通讯作者：
A. Champneys
中科院分区：
文献类型：
--
作者：
A. Champneys
如果存在与流动方向相反的相空间的对合，则称一个动力系统是可逆的。具有二次动能的经典哈密顿系统就是一个例子。对于可逆系统，在可逆变换下不变的同宿轨道通常随着参数的变化而保持不变。本文研究一类可逆系统，假设其存在一个到鞍点的主同宿轨道。正在研究的问题是关于次级同宿轨道的刻画，这些轨道存在于主轨道的邻域中。这类轨道可用作孤立水波和非线性支柱的屈曲解。对四维线性可逆系统进行了Shil‘nikov型分析。证明了每个辅助同宿轨道都可以用一个对称的正整数串来标号。所有长度为1、2或3的可能的弦都对应于同宿轨道的存在，而只有长度为4或更长的弦中的某些弦对应于同宿轨道的存在。如果可逆系统也是哈密顿的，则这种情况与已知结果形成对比。对P和α为参数的方程进行了仔细的数值实验，得到了与理论一致的结果。
A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are classical Hamiltonian systems with quadratic kinetic energy. For reversible systems, homoclinic orbits that are invariant under the reversible transformation typically persist as parameters are varied. This paper concerns reversible systems for which a primary homoclinic orbit to a saddle-focus is assumed to exist. The problem under investigation is a characterisation of the subsidiary homoclinic orbits which then exist in a neighbourhood of the primary one. Such orbits have applications as solitary water waves and as buckling solutions of nonlinear struts. A Shil’nikov-type analysis is performed for four-dimensional linearly reversible systems. It is shown that each subsidiary homoclinic orbit can be labelled by a symmetric string of positive integers. All possible strings of length one, two or three correspond to the existence of a homoclinic orbit, whereas only certain of those of length four or greater do. This situation contrasts with known results if the reversible system is also Hamiltonian. The analysis is supported by performing careful numerical experiments on the equation where P and α are parameters; a good agreement with the theory is found.