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Computing complete Lyapunov functions for discrete-time dynamical systems

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posted on 2023-06-09, 21:51 authored by Peter GieslPeter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein
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 flow 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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Discrete and Continuous Dynamical Systems Series B

ISSN

1531-3492

Publisher

American Institute of Mathematical Sciences

Issue

1

Volume

26

Page range

299-336

Pages

32.0

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2020-10-12

First Open Access (FOA) Date

2022-01-02

First Compliant Deposit (FCD) Date

2020-10-09

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