Efficient computation of Lyapunov functions for Morse decompositions
Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti · Discrete and Continuous Dynamical Systems - B · 2015
We present an efficient algorithm for constructing piecewise constant Lyapunov functions for dynamics generated by a continuous nonlinear map defined on a compact metric space.We provide a memory efficient data structure for storing nonuniform grids on which the Lyapunov function is defined and give bounds on the complexity of the algorithm for both time and memory.We prove that if the diameters of the grid elements go to zero, then the sequence of piecewise constant Lyapunov functions generated by our algorithm converge to a continuous Lyapunov function for the dynamics generated the nonlinear map.We conclude by applying these techniques to two problems from population biology.