Constructing the Lyapunov Function by Quasi-interpolation

Abstract

The Lyapunov function plays a vital role in dynamical systems. One method of constructing the Lyapunov function is the meshless collocation by radial basis functions. However, the meshless collocation method gives a non-local Lyapunov function (with unwanted nonnegative orbital derivative in some neighborhood of an equilibrium). To overcome the difficulty of the method, the paper proposes a scheme for constructing the Lyapunov function by multiquadric trigonometric B-spline quasi-interpolation. The scheme is efficient, simple and easy to compute. More importantly, it provides two additional shape parameters for the constructed Lyapunov function. This implies that, by adjusting these two shape parameters suitably, one can get different shapes of the Lyapunov function and thus obtain different basins of attraction for a dynamical system. Moreover, by taking the union of these different basins, one can even obtain a larger basin of attraction that provides better approximation to the exact one.