Lyapunov exponents and persistence in discrete dynamical systems

Paul L. Salceanu, Hal Smith

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

The theory of Lyapunov exponents and methods from ergodic theory have been employed by several authors in order to study persistence properties of dynamical systems generated by ODEs or by maps. Here we derive sufficient conditions for uniform persistence, formulated in the language of Lyapunov exponents, for a large class of dissipative discrete-time dynamical systems on the positive orthant of Rm, having the property that a nontrivial compact invariant set exists on a bounding hyperplane. We require that all so-called normal Lyapunov exponents be positive on such invariant sets. We apply the results to a plant-herbivore model, showing that both plant and herbivore persist, and to a model of a fungal disease in a stage-structured host, showing that the host persists and the disease is endemic.

Original languageEnglish (US)
Pages (from-to)187-203
Number of pages17
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume12
Issue number1
DOIs
StatePublished - Jul 2009

Fingerprint

Discrete Dynamical Systems
Lyapunov Exponent
Persistence
Dynamical systems
Invariant Set
Uniform Persistence
Dissipative Dynamical System
Discrete-time Dynamical Systems
Stage-structured
Lyapunov Methods
Ergodic Theory
Compact Set
Hyperplane
Dynamical system
Sufficient Conditions
Model

Keywords

  • Lyapunov exponents
  • Persistence
  • Uniformly weak repeller

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Lyapunov exponents and persistence in discrete dynamical systems. / Salceanu, Paul L.; Smith, Hal.

In: Discrete and Continuous Dynamical Systems - Series B, Vol. 12, No. 1, 07.2009, p. 187-203.

Research output: Contribution to journalArticle

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