Simple flows on tori with uncommon chaos
type: Item
instance of: scholarly article
title: Simple flows on tori with uncommon chaos
zbMATH Open document ID: 1450.37040
DOI: 10.1134/S1560354720020057
author: Carles Simó
published in: Regular and Chaotic Dynamics
publication date: 2020-04-14T00:00:00Z
review text: The author studies flows \(\varphi_t:\mathbb{T}^n\longrightarrow\mathbb{T}^n\) on the \(n\)-dimensional torus induced by the family of ODE's \[ \frac{d\mathbf{x}}{dt}=\Omega f_{\gamma}(\mathbf{x}), \] where \(f_{\gamma}(\mathbf{x})=f(\mathbf{x})^{\gamma/2}\) with \(f(\mathbf{x})=\sum_{i=1}^n f_i(x_i)\) and \(f_i(x_i)=\sin^2(\pi x_i)\) and \(\Omega=(\omega_1,\omega_2,\ldots,\omega_n)^T\) is a normalized vector whose components are positive and satisfy \(\omega_i<\omega_{j}\) for \(i< j\) and a nonresonance condition, namely \((\mathbf{k},\Omega)\neq 0\) for all \(\mathbf{k}\in\mathbb{Z}^n\setminus\{0\}\). The flows induced by these equations have a unique fixed point of parabolic type and neither homoclinic nor heteroclinic points. However, as it is shown in Theorem 1, these flows are {chaotic} in the sense that they are topologically transitive and present sensitive dependence on initial conditions. The author also studies bounds for the Lyapunov exponents in the case \(\gamma>1\). The main result in this direction is Theorem 2 that states that the maximal Lyapunov exponent whenever \(\gamma>1\) and \(\Omega\) is nonresonant is zero. The case \(\gamma=1\) is also treated for the \(2\)-dimensional torus showing that there are choices of \(\Omega\) that ensure that the maximal Lyapunov exponent is positive. In addition, some examples of maps with parabolic points and positive Lyapunov exponent are presented.
reviewed by: Héctor Barge
Mathematics Subject Classification ID: 37E35
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 7189264
zbMATH Keywords: chaos
zbMATH Keywords: homoclinic/heteroclinic points
zbMATH Keywords: chaotic flows on tori
zbMATH Keywords: quasi-periodicity
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: exceptional frequencies
MaRDI profile type: Publication
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full work available at URL: s1560354720020057
OpenAlex ID: W3015677927
description: scientific article; zbMATH DE number 7189264