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Multiplicity of homoclinic orbits for a class of asymptotically periodic Hamiltonian systems. (English) Zbl 0802.34052
Let us consider the asymptotically periodic Hamiltonian system (1) $$\ddot q= q-\text{grad }V(t,q)$$, $$q\in \mathbb{R}^ m$$, that means, $$U(t,q)$$ exists such that $$U(t+ T,q)= U(t,q)$$ for all $$(t,q)\in \mathbb{R}\times \mathbb{R}^ m$$ and $$|\text{grad }V(t,q)- \text{grad }U(t,q)|\to 0$$ as $$t\to- \infty$$, uniformly on the compacts of $$\mathbb{R}^ m$$. Under certain conditions, the author proved the existence of infinitely many homoclinic solutions of (1). The obtained solutions are so-called $$k$$-bump solutions $$k\in N$$, that means the $$k$$-bump solutions go away from zero and return near it, $$k$$ times.
Reviewer: A.Klíč (Praha)

MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58E30 Variational principles in infinite-dimensional spaces
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