Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems
Let \(T\) be a symmetric invertible real matrix which is indefinite, and let \(V\in C^2(\mathbb{R}^2, \mathbb{R})\) satisfy \(V(0)= 0\) and \(\nabla V(0)= 0\). Consider the equation (1) \(Tq''(t)+ \nabla V(q(t))= 0\), \(q\in C^2(\mathbb{R}, \mathbb{R}^2)\), and suppose that \(0\) is a saddle-focus, i.e. the eigenvalues \(\lambda\) of \(\lambda^2 Tq+ V''(0) q= 0\), \(q\in \mathbb{R}^2\), are complex (neither real, nor imaginary). In the present paper, the author proves under some geometrical conditions on \(V_0= \{q: V(q)= 0\}\) that there exist infinitely many homoclinic solutions of (1) and they may be found as limits of periodic solutions.
scientific article; zbMATH DE number 800403