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Remarks on dynamical systems with weak forces. (English) Zbl 0606.58039
In this paper the existence of T-periodic solutions for the forced system of differential equations -ÿ\(=\nabla V(y)+h(t)\) is studied, where h is T-periodic, V is defined in an open, bounded and convex subset \(\Omega \subset {\mathbb{R}}^ N\) and \(V\to -\infty\) as y tends to the boundary of the set \(\Omega\). The existence of at least one solution is proved, requiring convexity of -V near the boundary. Such a solution is found as a critical point of a functional \(\Phi\) defined via the dual action principle [see F. Clarke and I. Ekeland, Commun. Pure Appl. Math. 33, 103-116 (1980; Zbl 0403.70016)] and, for a situation very close to the one discussed here, see A. Ambrosetti and the author [Solutions with minimal period for Hamiltonian systems in a potential well, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (in print)]. The fact that the critical points of \(\Phi\) correspond to T-periodic solutions of class \(W^{2,1}\) is then exploited to prove that such a solution does not touch the boundary of \(\Omega\).

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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References:
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