Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system
The author proves that under certain assumptions (to complicate to be presented here) there exists \(\delta >0\) such that for sufficiently small \(\epsilon\) the following statement holds: for each real \(\alpha\) the system \({\dot \theta}=\epsilon f(\theta,\epsilon)\), \(\dot x=g(\theta,x)\) has an unique solution \(\theta(t)=\theta (t,\alpha,\epsilon)\), \(x(t,\alpha,\epsilon)\) satisfying \(\theta(0)=\alpha\), \(| x^{\pm}(t)-u^{\pm}(\theta (t))| \leq \delta\) for all t. Moreover \(\sup_{-\infty <t<\infty}| x^{\pm}(t,\alpha,\epsilon)- u^{\pm}(\theta (t,\alpha,\epsilon))| \to 0\) as \(\epsilon\to 0\) uniformly with respect to \(\alpha\) and the variational equations \(\dot x=g_ x(\theta (t,\alpha,\epsilon),\) \(x^{\pm}(t,\alpha,\epsilon))x\) has exponential dichotomies on \((-\infty,\infty)\). As an application the result generalizing the Chery's conjectured result is included. Here for each fixed \(\theta\) the system \(\dot x=g(\theta,x)\) has a saddle connection \(u(t,\theta)\) with associated saddle points \(u^-(t)\) and \(u^+(t)\) such that \(u_ t(t,\theta)\) is the unique bounded solution of the variational equation \(\dot x=g_ x(\theta,u(t,\theta)x).\)
scientific article; zbMATH DE number 3987717
scientific article