An Efficient Computational Framework for Studying Dynamical Systems

Abstract

In this paper, we introduce a computational framework for studying dynamical systems. This framework can be used to prove the existence of certain behaviour in a given dynamical system at any finite (limited) resolution automatically. The proposed framework is based on approximating the phase space topology of a given dynamical system at a finite resolution by adaptively partitioning it at rational points. Dyadic rationals and partition elements with disjoint interiors are employed to build a transparent partition that enables constructing an ideal combinatorial representation of a given dynamical system. Moreover, we introduce a new algorithmic strategy that overcomes the dependence on initial conditions, supports deriving ubiquitous conclusions, enables finding bifurcation points up to certain precision, and (most importantly) is computationally efficient. A set of simple yet powerful dynamic graph algorithms that were developed to support the new strategy are described in details. As an application, invariant sets and bifurcation points of the logistic map were computed.