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Looking for the Bernoulli shift. (English) Zbl 0803.58013
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. The main result is obtained by constructing multibump homoclinic solutions via variational methods.

MSC:
58E05Abstract critical point theory
58E30Variational principles on infinite-dimensional spaces
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
54C70Topological entropy
37D45Strange attractors, chaotic dynamics
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