Computing complete lyapunov functions for discrete-time dynamical systems

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein

Research output: Contribution to journalArticlepeer-review

Abstract

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the ow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction. We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.

Original languageEnglish
Pages (from-to)299-336
Number of pages38
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume26
Issue number1
DOIs
Publication statusPublished - 1 Jan 2021

Bibliographical note

Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.

Other keywords

  • Complete Lyapunov function
  • Discrete-time dynamical system
  • Meshfree collocation
  • Quadratic programming

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