Homoclinic orbits for second order Hamiltonian systems with potential changing sign.(English)Zbl 0867.70012

We study the second order Hamiltonian system $$\ddot q=-U'(t,q)$$, where $$q:\mathbb{R}\to \mathbb{R}^N$$ and $$U'(t,q)$$ denotes the gradient with respect to $$q$$ of a smooth potential $$U:\mathbb{R}\times \mathbb{R}^N\to\mathbb{R}$$, $$T$$-periodic in time, having an unstable equilibrium point $$\overline{x}$$ for all $$t\in\mathbb{R}$$. Without loss of generality we can take $$T=1$$ and $$\overline{x}=0$$. Thus, $$q(t)\equiv 0$$ is a trivial solution. We look for homoclinic orbits to 0, namely non-zero solutions of the problem $\ddot q=-U'(t,q), \qquad q(t)\to 0\text{ as }t\to\pm\infty, \qquad \dot q(t)\to 0\text{ as }t\to\pm\infty. \tag{P}$ The potential $$U$$ has the form $$U(t,x)=-{1\over 2}x\cdot L(t)x+V(t,x)$$, where $$L$$ and $$V$$ satisfy some technical assumptions. We prove that the problem (P) admits infinitely many solutions.

MSC:

 70H05 Hamilton’s equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems