Exponential dichotomies and transversal homoclinic orbits in degenerate cases
type: Item
instance of: scholarly article
title: Exponential dichotomies and transversal homoclinic orbits in degenerate cases
zbMATH Open document ID: 0844.34043
DOI: 10.1007/BF02218723
author: Weiyao Zeng
published in: Journal of Dynamics and Differential Equations
publication date: 1996-08-13T00:00:00Z
review text: Consider the system \(x' = A(t, \varepsilon) x\), \(x \in \mathbb{R}^n\), where \(\varepsilon \) is a small parameter. It is assumed that for \(\varepsilon = 0\) the system admits an exponential dichotomy both on \((- \infty,0]\) and on \([0, \infty)\). Assume that the system has \(d \geq 1\) linearly independent solutions bounded on \(\mathbb{R}\). The author gives conditions under which it admits exponential dichotomy on \(\mathbb{R}\) (this generalizes a well-known result of Palmer for \(d = 1)\). The result is applied to study the existence of transversal homoclinic orbits for the system \(x' = g(x) + \varepsilon h(t,x, \varepsilon)\) which has a homoclinic orbit \(r\) for \(\varepsilon = 0\). Sufficient conditions for the existence of transversal homoclinic orbits near \(r\) are given in the following two degenerate cases: (i) the stable and unstable manifolds intersect only at \(r\), but their tangent spaces have an intersection of dimension at least 2; (ii) the stable and unstable manifolds have a \(d\)-dimensional family of homoclinic orbits that contains \(r\) and \(d \geq 2\).
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 851867
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: homoclinic orbit
reviewed by: Sergei Yu. Pilyugin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 851867