A natural order in dynamical systems based on Conley–Markov matrices

We introduce a natural order to study properties of dynamical systems, especially their invariant sets. The new concept is based on the classical Conley index theory and transition probabilities among neighborhoods of different invariant sets when the dynamical systems are perturbed by white noises. The transition probabilities can be determined by the Fokker–Planck equation and they form a matrix called a Markov matrix. In the limiting case when the random perturbation is reduced to zero, the Markov matrix recovers the information given by the Conley connection matrix. The Markov matrix also produces a natural order from the least to the most stable invariant sets for general dynamical systems. In particular, it gives the order among the local extreme points if the dynamical system is a gradient-like flow of an energy functional. Consequently, the natural order can be used to determine the global minima for gradient-like systems. Some numerical examples are given to illustrate the Markov matrix and its properties.