# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 58E05 Abstract critical point theory 58E30 Variational principles on infinite-dimensional spaces 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 54C70 Topological entropy 37D45 Strange attractors, chaotic dynamics
Full Text: