Comparing dynamical systems by a graph matching method

• We compare dynamical systems via a graph matching method.
• We prove that for conjugate systems, the matching matrix coincides with the solution of the associated Monge’s problem.
• We give an example and use EMD to show the matching matrix.
• We give an error analysis and regularity analysis for our method.
• For one system which embeds into the other, we prove that the comparing is a subgraph matching.

In this paper, we consider comparing dynamical systems by using a method of graph matching, either between the graphs representing the underlying symbolic dynamics, or between the graphs approximating the action of the systems on a fine but otherwise non-generating partition. For conjugate systems, the graphs are isomorphic and we show that the permutation matrices that relate the adjacency matrices coincide with the solution of Monge’s mass transport problem. We use the underlying earth mover’s distance (EMD) to generate the “approximate” matching matrix to illustrate the association of graphs which are derived from equal-distance partitioning of the phase spaces of systems. In addition, for one system which embeds into the other, we show that the comparison of these two systems by our method is an issue of subgraph matching.