Exponential dichotomies and transversal homoclinic orbits in degenerate cases
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 \(d \geq 2\).
scientific article; zbMATH DE number 851867
scientific article