Algorithmic verification of approximations to complete Lyapunov functions

In [3] Conley showed that the state-space of a dynamical system can be decomposed into a gradient-like part and a chain-recurrent part, and that this decomposition is characterized by a so-called complete Lyapunov function for the system. In [14] Kalies, Mischaikow, and VanderVorst proposed a combinatorial method to compute discrete approximations to such complete Lyapunov functions. Their approach uses a finite subdivision of a compact subset of the state-space and a combinatorial multivalued map to approximate the dynamics. They proved that as the diameter of the elements of the subdivision approaches zero the resulting approximations to complete Lyapunov functions converge to a true complete Lyapunov function for the system. In [2] Ban and Kalies implemented this algorithm and used it to compute an approximation to a complete Lyapunov function for a time-T mapping of the van der Pol oscillator. The CPA method to compute Lyapunov functions uses linear programming to parameterize true Continuous and Piecewise Affine Lyapunov functions for continuous dynamical systems [11], [1], [8]. In this paper we propose using the CPA method to evaluate approximations to complete Lyapunov functions computed by the combinatorial method. Especially, we can explicitly compute the region of the state-space where the orbital derivative of the approximation is negative. Further, we use the RBF method [5] to solve a Zubov equation ∇V (x) · f(x) = −p(x) and compare the solution V with the complete Lyapunov function computed by the combinatorial method for the van der Pol oscillator.