Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems

In this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system. These methods are based on the concept of a maximal Lyapunov function, which is introduced in this paper. A partial differential equation characterizing a maximal Lyapunov function is derived, and the relationships of this equation as compared to Zubov's partial differential equation are discussed. An iterative procedure is given for solving the new partial differential equation. This procedure yields Lyapunov function candidates that are rational functions rather than polynomials. The method is applied to four two-dimensional examples and one three-dimensional example, and it is shown that the estimates obtained using this method are, in many cases, substantially better than those obtained using known methods.