2015/08/10 by Gedeon, Tomas, Harker, Shaun, Kokubu, Hiroshi +2
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1508.02438
We introduce a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. Motivated by models of regulatory networks, we construct a state transition graph from a piecewise affine ordinary differential equation. We use efficient graph algorithms to compute an associated Morse graph that codifies the recurrent and gradient-like dynamics. We prove that for 2-dimensional systems, the Morse graph defines a Morse decomposition for the dynamics of any smooth differential equation that is sufficiently close to the original piecewise affine ordinary differential equation.