Covering relations for multidimensional dynamical systems
type: Item
instance of: scholarly article
title: Covering relations for multidimensional dynamical systems
zbMATH Open document ID: 1061.37013
author: Marian Gidea
author: Piotr Zgliczyński
published in: Journal of Differential Equations
publication date: 2004-08-19T00:00:00Z
review text: A form of topological hyperbolicity is introduced. It gives a sufficient condition for positive topological entropy (one form of ``chaos'') and existence of infinitely many periodic orbits. This condition can be checked by finite precision computations. It is a generalization, in the presence of contracting behavior, of the basic fact that if \(f\) is a continuous map and \(B_1,\dots, B_n,B_{n+1}:=B_n\) are compact topological balls with \(f(B_{i})\supset B_{i+1}\), then there exists \(x\in B_1\) with \(f^n(x)=x\) and \(f^{i-1}(x)\in B_i\) for \(1\leq i\leq n\). Let us state the condition replacing \(f(B_i)\supset B_{i+1}\). \(B_i,B_{i+1}\) are identified with standard rectangles \(I_u\times I_s\) (\(I_u,I_s\) are disks in Euclidean spaces) such that, in this identification, the map \(f| B_i\) is homotopic to an affine map with image \(\Delta\) satisfying \(\{0\}\times I_u\subset\Delta\subset\{0\}\times \hat I_u\) with \(\hat I_u\) a larger disk. Additionally, the homotopy must respect boundaries in the following sense: it does not intersect \(I_u\times\partial I_s\) and it maps \(I_s\times\partial I_u\) away from \(I_u\times I_s\).
reviewed by: Jérôme Buzzi
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 2092983
zbMATH Keywords: fixed point index
zbMATH Keywords: periodic points
zbMATH Keywords: chaos detection
zbMATH Keywords: topological entropy
MaRDI profile type: Publication
full work available at URL: j.jde.2004.03.013
OpenAlex ID: W2048280869
cites work: Periodic orbits, symbolic dynamics and topological entropy for the restricted 3-body problem
cites work: Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level
cites work: Q3941926
cites work: Isolating blocks and symbolic dynamics
cites work: Homoclinic phenomena in Hamiltonian systems with several degrees of freedom
cites work: Positive Topological Entropy of Chua's Circuit: A Computer Assisted Proof
cites work: Computer assisted proof of chaos in the Lorenz equations
cites work: A Chaos Lemma
cites work: Topological horseshoes
cites work: A generalized Lefschetz fixed point theorem and symbolic dynamics in delay equations
cites work: Isolating neighborhoods and chaos
cites work: Q4243438
cites work: TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS
cites work: Q5601742
cites work: A generalization of the Lefschetz fixed point theorem and detection of chaos
cites work: Isolating segments, fixed point index, and symbolic dynamics
cites work: Heteroclinic connections between periodic orbits in planar restricted circular three-body problem -- a computer-assisted proof
cites work: Fixed point index for iterations of maps, topological horseshoe and chaos
cites work: Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps
cites work: On periodic points for systems of weakly coupled 1-dim maps.
cites work: Computer assisted proof of chaos in the Rössler equations and in the Hénon map
cites work: MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS AND STABILITY OF SARKOVSKĬ ORDERING
DOI: 10.1016/J.JDE.2004.03.013
description: scientific article; zbMATH DE number 2092983