Making a dynamical system chaotic: feedback control of Lyapunov exponents for discrete-time dynamical systems

1997 ◽  
Vol 44 (3) ◽  
pp. 250-253 ◽  
Author(s):  
Guanrong Chen ◽  
Dejian Lai
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Exponents ◽  
Discrete Time ◽  
Discrete Time Dynamical Systems

scholarly journals Lyapunov maps, simplicial complexes and the Stone functor

1992 ◽  
Vol 12 (1) ◽  
pp. 153-183 ◽  
Author(s):  
Joel W. Robbin ◽  
Dietmar A. Salamon
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Simplicial Complexes ◽  
Attractor Network ◽  
Partially Ordered ◽  
Connection Matrices ◽  
Discrete Time Dynamical Systems

AbstractLet be an attractor network for a dynamical system ft: M → M, indexed by the lower sets of a partially ordered set P. Our main theorem asserts the existence of a Lyapunov map ψ:M → K(P) which defines the attractor network. This result is used to prove the existence of connection matrices for discrete-time dynamical systems.

CHAOTIFICATION OF DISCRETE-TIME DYNAMICAL SYSTEMS: AN EXTENSION OF THE CHEN–LAI ALGORITHM

2005 ◽  
Vol 15 (01) ◽  
pp. 109-117 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Design Method ◽  
Control Design ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Discrete Time Dynamical Systems ◽  
Uniformly Bounded

In this article, we propose and study an extension of the Chen–Lai algorithm for chaotification of discrete-time dynamical systems. The proposed method is a simple but mathematically rigorous feedback control design method that can gradually make all the Lyapunov exponents of the controlled system strictly positive for any given n-dimensional dynamical system that has a uniformly bounded Jacobian but otherwise could be originally nonchaotic or even asymptotically stable.

FEEDBACK CONTROL OF LYAPUNOV EXPONENTS FOR DISCRETE-TIME DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1341-1349 ◽  
Author(s):  
GUANRONG CHEN ◽  
DEJIAN LAI
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Exponents ◽  
Design Method ◽  
Design Procedure ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Convenient Technique ◽  
Discrete Time Dynamical Systems

A simple, yet mathematically rigorous feedback control design method is proposed in this paper, which can make all the Lyapunov exponents of the controlled system strictly positive, for any given n-dimensional dynamical system that could be originally nonchaotic or even asymptotically stable. The argument used is purely algebraic and the design procedure is completely schematic, with no approximations used throughout the derivation. This is a rigorous and convenient technique suggested as an attempt for anticontrol of chaotic dynamical systems, with explicit computational formulas derived for applications.

CHAOTIC DYNAMICS FOR MAPS IN ONE AND TWO DIMENSIONS: A GEOMETRICAL METHOD AND APPLICATIONS TO ECONOMICS

2009 ◽  
Vol 19 (10) ◽  
pp. 3283-3309 ◽  
Author(s):  
ALFREDO MEDIO ◽  
MARINA PIREDDU ◽  
FABIO ZANOLIN
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Dynamics ◽  
Economic Models ◽  
Two Dimensions ◽  
Geometrical Method ◽  
Mathematical Ideas ◽  
Definition Of ◽  
Applications To Economics ◽  
Discrete Time Dynamical Systems

This article describes a method — called here "the method of Stretching Along the Paths" (SAP) — to prove the existence of chaotic sets in discrete-time dynamical systems. The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications. The paper provides a description of the basic mathematical ideas behind the method, as well as three applications to economic models. Incidentally, the paper also discusses some questions concerning the definition of chaos and some problems arising from economic models in which the dynamics are defined only implicitly.

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