Lyapunov exponents and persistence in discrete dynamical systems

Paul L. Salceanu, Hal Smith

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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
Issue number1
StatePublished - Jul 2009


  • Lyapunov exponents
  • Persistence
  • Uniformly weak repeller

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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