Existence of homoclinic solutions to periodic orbits in a center manifold
The authors study a Hamiltonian system with Lagrangian \[ L(x,\dot{x},q,\dot{q})=\tfrac{1}{2}(\dot{x}^2 - x^2)+ \tfrac{1}{2}\dot{q}^2 + (1+\delta(x))V(q), \] where \(V\) is \(2\pi\)-periodic, nonnegative and \(V''(0)=0\) while \(\delta\) and \(\delta'\) satisfy some boundedness conditions. The point \(q=x=0\) is a saddle-center equilibrium with a two-dimensional center manifold \(\{q=\dot{q}=0\}\) foliated by periodic orbits. The main result states that if there is no homoclinic orbit to the equilibrium, then there exists an orbit which is homoclinic to some periodic orbit. In addition, the phase difference of the asymptotic phases at \(t=-\infty\) and \(t=+\infty\) is contained in an arbitrary interval \([\alpha,\beta]\subseteq [0,2\pi]\). In particular, this implies the existence of infinitely many homoclinic orbits to periodic orbits in the center manifold. The proof is based on variational techniques: Using the Palais-Smale property of some suitable functional, periodic solutions with large period are constructed. The homoclinic orbits are then found as limits of such periodic orbits.
scientific article; zbMATH DE number 2092988