Subharmonic solutions of second order subquadratic Hamiltonian systems with potential changing sign
This paper considers the existence of periodic solutions of a system of equations \(-\ddot{x} = V_x (t,x)\) for \(x \in {\mathbb R}^n\). The equations are written in Hamiltonian form as \(-\dot{x} = H_y (t,x,y)\), \(\dot{y} = H_x (t,x,y)\) where \(H(t,x,y) = \frac{1}{2} |y|^2 + V(t,x)\), and where the potential \(V(t,x)\) is subquadratic and may change sign. The author considers systems where \(V = V_1 + V_2\) with both \(V_1\) and \(V_2\) 1-periodic, \(V_1\) positive at infinity, and \(V_2\) satisfying \(\int_0^1 V_{2,x} (t,x) dt = 0\), and combines both time-dependent and time-independent symplectic transformations to construct a new Hamiltonian independent of the sign-changing potential \(V_2\) so that known existence results can be used. Conditions are given for the functions \(V_1\) and \(V_2\) that guarantee the existence of \(k\)-periodic solutions \(x_k\) such that \(\text{ max }_t (|x_k(t)|+ |\dot{x}_k(t)|) \rightarrow \infty\) as \(k \rightarrow \infty\). In the case where \(x \in {\mathbb R}\), the author also gives some additional conditions on \(V\) to guarantee the existence of Aubrey-Mather sets for the system.
scientific article; zbMATH DE number 1449208