Computing complete Lyapunov functions for discrete-time dynamical systems

Giesl, Peter, Langhorne, Zachary, Argáez, Carlos and Hafstein, Sigurdur (2021) Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete and Continuous Dynamical Systems Series B, 26 (1). pp. 299-336. ISSN 1531-3492

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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 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.

Item Type: Article
Keywords: Discrete-time dynamical system, Complete Lyapunov function, Quadratic programming
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 12 Oct 2020 10:36
Last Modified: 18 Jan 2021 13:30
URI: http://sro.sussex.ac.uk/id/eprint/94284

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