Construction of Lyapunov functions for the estimation of basins of attraction

Abstract

Technical systems are often modeled through systems of differential equations in which the parameters and initial conditions are subject to uncertainties. Usually, special solutions of the differential equations like equilibrium positions and periodic orbits are of importance and frequently the corresponding equations are only set up with the intent to describe the behavior in the vicinity of a limit cycle or an equilibrium position. For the validity of the analysis it must therefore be assumed that the initial conditions lie indeed in the basins of attraction of the corresponding attractors. In order to estimate basins of attraction, Lyapunov functions can be used. However, there are no systematic approaches available for the construction of Lyapunov functions with the goal to achieve a good approximation of the basin of attraction. The present paper suggests a method for defining appropriate Lyapunov functions using insight from center manifold theory. With this approach, not only variations in the initial conditions, but also in the parameters can be studied. The results are used to calculate the likelihood for the system to reach a certain attractor assuming different random distributions for the initial conditions.