×

Looking for the Bernoulli shift. (English) Zbl 0803.58013

Consider a Hamiltonian system \(-J \dot x = A x + R_ x(t,x)\), where \(J\) is the canonical symplectic matrix, \(A\) is Hermitian, and \(JA\) is a constant matrix with nonzero real parts of all its eigenvalues. Assume that \(R\) is 1-periodic in \(t\), and is strictly convex \(\forall t\), and that for some \(\alpha > 0\), \(0 < k_ 1 < k_ 2 < +\infty\), we have \[ k_ 1 | x|^ \alpha \leq R(t,x) \leq k_ 2| x|^ \alpha. \] Suppose that the set of nonzero critical points of the dual action functional associated with the system is at most countable below the level \(c^ 1> c\), where \(c\) is the mountain pass level. Then there exists a homoclinic orbit \(x\) such that, for any \(\varepsilon > 0\) and any \(\overline{p} = (p^ 1,\dots ,p^ m) \in \mathbb{Z}^ m\) satisfying \[ \forall i : (p^{i + 1} - p^ i) \geq K(\varepsilon),\quad\text{a const. independent of }m, \] there is a homoclinic orbit \(y_{\overline{p}}\) with \[ \biggl\| y_{\overline{p}} - \sum^ m_{i = 1} x( \cdot - p^ i) \biggr\|_ \infty \leq \varepsilon. \] As a consequence, the flow of the system has a positive topological entropy.
The main result is obtained by constructing multibump homoclinic solutions via variational methods.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
54C70 Entropy in general topology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[3] Coti-Zelati, V.; Ekeland, I.; Séré, E., A Variational Approach to Homoclinic Orbits in Hamiltonian Systems, Mathematische Annalen, Vol. 288, 133-160 (1990) · Zbl 0731.34050
[6] Ekeland, I., Convexity Methods in Hamiltonian Systems (1989), Springer Verlag
[7] Hofer, H.; Wysocki, K., First Order Elliptic Systems and the Existence of Homoclinic Orbits in Hamiltonian Systems, Math. Annalen, Vol. 288, 483-503 (1990) · Zbl 0702.34039
[10] Lions, P. L., The Concentration-Compactness Principle in the Calculus of Variations, Revista Iberoamericana, Vol. 1, 145-201 (1985) · Zbl 0704.49005
[11] Moser, J., Stable and Random Motions in Dynamical Systems (1973), Princeton University Press: Princeton University Press Princeton · Zbl 0271.70009
[13] Séré, E., Existence of Infinitely Many Homoclinic Orbits in Hamiltonian Systems, Math. Zeitschrift, Vol. 209, 27-42 (1992) · Zbl 0725.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.