Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems
The existence of infinitely many homoclinic solutions [i.e., solutions \(q\) such that \(q(t)\) and \(q'(t)\) tend to the rest point \(q=0\) as \(t\to\pm\infty]\) for a certain class of second-order Hamiltonian systems in \(\mathbb{R}^N\) is proved. As mentioned by the authors, the existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behaviour of dynamical systems was already recognized by Poincaré, but only since fifteen years ago variational methods have been employed to deal with this issue [\textit{S. Bolotin}, Mosc. Univ. Math. Bull. 38, No. 6, 117-123 (1983); translation from Vestn. Mosk. Univ., Ser. I 1983, No. 6, 98-103 (1983; Zbl 0549.58019)]. More recently, \textit{E. Séré} proved the existence of a class of solutions -- called multi-bump solutions -- for a certain class of first-order, convex, periodically forced Hamiltonian systems [Math. Z. 209, No. 1, 27-42 (1992; Zbl 0739.58023); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No. 5, 561-590 (1993; Zbl 0803.58013)]. Notably, the existence of this class of solutions implies that the dynamics of the system are chaotic (in particular, its topological entropy is positive). In order to obtain this result a nondegeneracy condition -- verified when the set of homoclinic solutions is countable -- had to be imposed. Such results have been later extended to the almost periodic case and to singular Hamiltonian systems in \(\mathbb{R}^2\). In the present work, a new theorem on the existence of infinitely many homoclinic solutions is proved for a class of Hamiltonian systems in \(\mathbb{R}^N\) of the form \(-\ddot q= -q+\alpha(t)W'(q)\), under the hypothesis that the function \(\alpha(t)\) is almost periodic and \(W(q)\) superquadratic. It is also shown that this system has multi-bump homoclinic solutions whenever a suitable non-degeneracy condition is satisfied. As remarked by the authors themselves, their theorem is significant when \(\alpha\) is not a constant function, and is not periodic in time.
scientific article; zbMATH DE number 993524