An Algorithmic Approach to Lattices and Order in Dynamics

Recurrent versus gradient-like behavior in global dynamics can be characterized via a surjective lattice homomorphism between certain bounded, distributive lattices, that is, between attracting blocks (or neighborhoods) and attractors. Using this characterization, we build finite, combinatorial models in terms of surjective lattice homomorphisms, which lay a foundation for a computational theory for dynamical systems that focuses on Morse decompositions and index lattices. In particular we present an algorithm that builds a combinatorial model that represents a Morse decomposition for the underlying dynamics. We give computational examples that illustrate the theory for both maps and flows.