Looking for the Bernoulli shift
type: Item
instance of: scholarly article
title: Looking for the Bernoulli shift
zbMATH Open document ID: 0803.58013
DOI: 10.1016/S0294-1449(16)30205-0
author: Éric Séré
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1995-01-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78317
review text: 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.
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 554347
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: topological entropy
zbMATH Keywords: variational methods
reviewed by: Kung Ching Chang
MaRDI profile type: Publication
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OpenAlex ID: W2260911434
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description: scientific article; zbMATH DE number 554347