Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems

1992 ◽  
Vol 116 (4) ◽  
pp. 339-360 ◽  
Author(s):  
P. Poláčik ◽  
I. Tereščák
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Strongly Monotone ◽  
Discrete Time Dynamical Systems

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.

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