Isolating segments, fixed point index, and symbolic dynamics
type: Item
instance of: scholarly article
title: Isolating segments, fixed point index, and symbolic dynamics
zbMATH Open document ID: 0953.34034
published in: Journal of Differential Equations
publication date: 2001-01-15T00:00:00Z
review text: The authors extend the method of \textit{R. Srzednicki} and \textit{K. Wójcik} [J. Differ. Equations 135, No.~1, 66-82 (1997; Zbl 0873.58049)] for the detection of chaotic dynamics in periodically forced ordinary differential equations. Their method is based on checking the so-called topological hyperbolicity by construction of periodic isolating segments. As an example, they show that the following planar system (written in the complex form) \[ z'=\left(1+e^{ikt}|z|^2\right)z \] has a symbolic dynamics on three symbols for \(0<k<0.495\).
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 1437327
zbMATH Keywords: chaotic dynamics
zbMATH Keywords: periodically forced ordinary differential equations
zbMATH Keywords: topological hyperbolicity
author: Klaudiusz Wójcik
author: Piotr Zgliczyński
reviewed by: Sergei Yu. Pilyugin
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3696
OpenAlex ID: W2077510959
cites work: Q4180600
cites work: Q5643279
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Computer assisted proof of chaos in the Lorenz equations
cites work: Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations
cites work: On detection of chaotic dynamics in ordinary differential equations
cites work: Q4347840
cites work: A geometric method for detecting chaotic dynamics
cites work: Isolating segments and symbolic dynamics
cites work: Fixed point index for iterations of maps, topological horseshoe and chaos
cites work: Computer assisted proof of chaos in the Rössler equations and in the Hénon map
DOI: 10.1006/JDEQ.1999.3696
description: scientific article; zbMATH DE number 1437327