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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
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