Introduction to Discrete Chaotic Dynamical Systems

If you ever wondered about the meaning and purpose of basins of attraction, systems with bifurcations, the universal constant of chaos, the transfer operator and the related Frobenius-Perron framework, the Lyapunov exponent, fractal dimensions and fractional Brownian motions, or how to measure and synthetize chaos, you will find the answer in this chapter. Even with a short, simple mathematical proof on occasion, but definitely at a level accessible to first year college students, with focus on examples. The chaotic systems described here are used in various applications and typically taught in advanced classes. I hope that my presentation makes this beautiful theory accessible to a much larger audience.

Many more systems (typically called maps or mappings) will be described in the next chapters. But even in this introductory material, you will be exposed to the Gauss map and its relation to generalized continued fractions, bivariate numeration systems, attractors, the 2D sine map renamed “pillow map” based on the above picture, systems with exact solution in closed form, a curious excellent approximation of π based on the first digit in one particular system, non-integer bases, digits randomization, and how to compute the invariant probability distribution. The latter is usually called invariant measure, but I do not make references to advanced measure theory in this book.

To read more, access the Python code and download the 17-pages article (chapter 2 of my upcoming book),?follow this link.