Overview of Empirical Distribution Function Computational Framework to Explore Chaotic Dynamic Systems

This paper describes a working methodology of implementation of empirical distribution functions used as a powerful tool for a theoretical exploration and a visualization of chaotic dynamic systems. To illustrate the framework, a dynamic process generated by a generalized logistic map called LM(1/2,1/2, 1) is considered. The dynamic sequence is represented as a stochastic process with invariant distributions. Mathematical equations for invariant transformations are derived and numerically solved. It is shown that the algorithm may be very sensitive to the size of the computational grid. The paper also describes a computational procedure for identifying unstable periodic orbits and demonstrates convergence of their distributions to real distribution functions of dynamic regimes. Such behavior is postulated as a property of chaos.