Arnold tongues in the problem of large-amplitude periodic trajectories
type: Item
instance of: scholarly article
title: Arnold tongues in the problem of large-amplitude periodic trajectories
zbMATH Open document ID: 1205.37065
DOI: 10.1134/S106456240606010X
published in: Doklady Mathematics
publication date: 2010-12-08T00:00:00Z
review text: Consider a finite-dimensional discrete time dynamical system \(x_{k+1}=U_{\lambda}x_k\), \(x\in {\mathbb R}^n\) depending on a complex parameter \(\lambda\). In a neighbourhood of an equilibrium (say, at \(x_*=0\)) its properties are determined by its linear part \(A_{\lambda}x\). If the matrix \(A_{\lambda}\) has eigenvalues near the unit circle, then in a neighbourhood of \(0\) the system may have periodic orbits with different periods. Periodic regimes occur for eigenvalues of the matrix \(A_{\lambda}\) from certain domains \(U_q\) (with vertices at the points \(e^{2\pi qi}\)) called Arnold tongues, where \(q\in {\mathbb Q}\). The paper studies such tongues at infinity, assuming that the operator \(U_{\lambda}\) is of the form \(A_{\lambda}x+\)bounded nonlinearity.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5824751
zbMATH Keywords: Arnold tongue
zbMATH Keywords: finite-dimensional discrete time dynamical system
zbMATH Keywords: periodic regime
author: Victor Kozyakin
author: Alexander Krasnosel'skij
author: Dmitrii Rachinskii
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
cites work: Elements of applied bifurcation theory.
full work available at URL: s106456240606010x
OpenAlex ID: W1981239281
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description: scientific article; zbMATH DE number 5824751