Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

Abstract

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x n+1 = g(x n ) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523-546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.

Variable Gradient Approach to Construct Lyapunov Function for Judging Stability of Non-Linear Systems

“…For the non-linear systems, it is still dificult to analyze the stability even though the Lyapunov methods are used. There has to find a positive definite Lyapunov function and its derivative function has to be negative definite [3,4,5].…”

Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

“…Numerical methods to compute Lyapunov functions for nonlinear discrete systems have, for example, been presented in [11,12], where collocation is used to solve numerically a discrete analog to Zubov s partial differential equation [ ] using radial basis functions [8,40] and in [4,23], where graph algorithms are used to compute complete Lyapunov functions [9,35]. For nonlinear systems with a certain structure there are many more approaches in the literature.…”