  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.
