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Zbl 0803.58013
Séré, Éric
Looking for the Bernoulli shift.
(English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No.5, 561-590 (1993). ISSN 0294-1449

Consider a Hamiltonian system $-J \dot x = A x + R\sb x(t,x)$, 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 $\alpha > 0$, $0 < k\sb 1 < k\sb 2 < +\infty$, we have $$k\sb 1 \vert x\vert\sp \alpha \leq R(t,x) \leq k\sb 2\vert x\vert\sp \alpha.$$ Suppose that the set of nonzero critical points of the dual action functional associated with the system is at most countable below the level $c\sp 1> c$, where $c$ is the mountain pass level. Then there exists a homoclinic orbit $x$ such that, for any $\varepsilon > 0$ and any $\overline{p} = (p\sp 1,\dots ,p\sp m) \in \bbfZ\sp m$ satisfying $$\forall i : (p\sp{i + 1} - p\sp i) \geq K(\varepsilon),\quad\text{a const. independent of }m,$$ there is a homoclinic orbit $y\sb{\overline{p}}$ with $$\biggl\Vert y\sb{\overline{p}} - \sum\sp m\sb{i = 1} x( \cdot - p\sp i) \biggr\Vert\sb \infty \leq \varepsilon.$$ As a consequence, the flow of the system has a positive topological entropy.\par The main result is obtained by constructing multibump homoclinic solutions via variational methods.
[Chang Kungching (Beijing)]
MSC 2000:
*58E05 Abstract critical point theory
58E30 Variational principles on infinite-dimensional spaces
37J99 Finite-dimensional Hamiltonian etc. systems
54C70 Topological entropy
37D45 Strange attractors, chaotic dynamics

Keywords: Hamiltonian systems; homoclinic orbits; topological entropy; variational methods

Cited in: Zbl 0878.34045

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