A Finite-time Condition for Exponential Trichotomy in Infinite Dynamical Systems

2015 ◽  
Vol 67 (5) ◽  
pp. 1065-1090 ◽  
Author(s):  
Arnaud Ducrot ◽  
Pierre Magal ◽  
Ousmane Seydi
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Finite Time ◽  
Time Condition ◽  
Infinite Dimensional ◽  
Hyperbolic Set ◽  
Exponential Trichotomy ◽  
Discrete Time Dynamical Systems

AbstractIn this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

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.

Large-Scale Discrete-Time Interconnected Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Large Scale ◽  
Discrete Energy ◽  
Flow Balance ◽  
Subsystem Level ◽  
Coupling Strengths ◽  
Definition Of ◽  
Energy Dissipated ◽  
Discrete Time Dynamical Systems

This chapter develops vector dissipativity notions for large-scale nonlinear discrete-time dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear discrete-time dynamical systems in terms of a vector dissipation inequality involving a vector supply rate, a vector storage function, and a nonnegative, semistable dissipation matrix. On the subsystem level, the proposed approach provides a discrete energy flow balance in terms of the stored subsystem energy, the supplied subsystem energy, the subsystem energy gained from all other subsystems independent of the subsystem coupling strengths, and the subsystem energy dissipated. The chapter also develops extended Kalman–Yakubovich–Popov conditions, in terms of the local subsystem dynamics and the interconnection constraints, for characterizing vector dissipativeness via vector storage functions for large-scale 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.

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