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The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign. (English) Zbl 0871.58036
The paper deals with second order Hamiltonian systems of the form $$q''-L(t)q+b(t)\text{grad}W(q)=0,$$ where $q\in\bbfR^n$, $L(t)$ is a symmetric positive definite matrix, and the function $b(t)$ can change sign. The aim is to find non-trivial homoclinic orbits through 0. The author proves three theorems giving the conditions for existence of these orbits. The existence of homoclinic orbits of such systems has been studied by many mathematicians, and the considered generalization concerns a new kind of “superquadratic” condition on $W$ and the assumption changing sign of function $b(t)$; in so doing $b(t)$ may be non-periodic.

37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34C25Periodic solutions of ODE