Multiplicity of homoclinic orbits for a class of asymptotically periodic Hamiltonian systems
Let us consider the asymptotically periodic Hamiltonian system (1) \(\ddot q= q-\text{grad }V(t,q)\), \(q\in \mathbb{R}^ m\), that means, \(U(t,q)\) exists such that \(U(t+ T,q)= U(t,q)\) for all \((t,q)\in \mathbb{R}\times \mathbb{R}^ m\) and \(|\text{grad }V(t,q)- \text{grad }U(t,q)|\to 0\) as \(t\to- \infty\), uniformly on the compacts of \(\mathbb{R}^ m\). Under certain conditions, the author proved the existence of infinitely many homoclinic solutions of (1). The obtained solutions are so-called \(k\)-bump solutions \(k\in N\), that means the \(k\)-bump solutions go away from zero and return near it, \(k\) times.
asymptotically periodic Hamiltonian system
scientific article; zbMATH DE number 591105