Computing complete Lyapunov functions for discrete-time dynamical systems

Peter Giesl, Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein, Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland
2017 Discrete and continuous dynamical systems. Series B  
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
more » ... ose 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. 2020 Mathematics Subject Classification. Primary: 93D30, 65D12, 65K10; Secondary: 39A30.
doi:10.3934/dcdsb.2020331 fatcat:uymicvegxjhnnme26fcnn4lrua