Attractors of control systems.
type: Item
instance of: scholarly article
title: Attractors of control systems.
zbMATH Open document ID: 1039.37077
author: N. A. Bobylev
author: S. V. Emel'yanov
author: S. K. Korovin
published in: Doklady Mathematics
publication date: 2004-06-15T00:00:00Z
review text: Let \({\mathcal M}\) be a metric compact space with a metric \(d(\cdot, \cdot)\), and let \(f_i:{\mathcal M}\to {\mathcal M}\), \(i=1,2,\dots,k\) be contraction maps. On \({\mathcal M}\), we consider the discrete dynamical system \[ x_{n+1}= f_{m(n)}(x_n), \quad n=0,1,\dots, \tag{1} \] where \[ m(\cdot):\mathbb{N} \cup\{0\} \to\{1,2, \dots,k\} \tag{2} \] is a function. In what follows, the function \(m(\cdot)\) involved in the definition of the discrete system (1) is treated as an open-loop control that, together with a chosen initial condition \(x_0\), generates the trajectory \(\{x_n\}\) of system (1). The set of all controls \(m(\cdot)\) defined by (2) is denoted by \({\mathfrak M}\). Here, we examine the qualitative behavior of trajectories of all discrete systems (1) rather than particular problems arising in the theory of discrete-system control. It turns out that each control \(m(\cdot) \in{\mathfrak M}\) is associated with a compact attractor \({\mathcal A}\subset {\mathcal M}\) of the corresponding discrete system, i.e., with a set to which all trajectories of this system approach. This attractor does not depend on the choice of \(m(\cdot)\) and is common for all systems (1). Moreover, for almost all controls \(m(\cdot)\in {\mathfrak M}\), the trajectories of the corresponding systems (1) are dense in \({\mathcal A}\).
Mathematics Subject Classification ID: 37N35
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 34K35
Mathematics Subject Classification ID: 93D05
zbMATH DE Number: 2074511
zbMATH Keywords: data compression and storage
zbMATH Keywords: fractal image processing
zbMATH Keywords: dynamical systems with delay
zbMATH Keywords: computer simulation
zbMATH Keywords: chaotic dynamics
zbMATH Keywords: discrete dynamical system
zbMATH Keywords: open-loop control
zbMATH Keywords: qualitative behavior of trajectories
zbMATH Keywords: attractor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2074511