Introduction to Discrete Chaotic Dynamical Systems

Entitled “Introduction to Discrete Chaotic Dynamical Systems”, the full version in PDF format is accessible in the “Free Books and Articles” section, here. This article is an extract from my book “Gentle Introduction to Chaotic Dynamical Systems”, available here.

This is chapter 2 of my upcoming book on dynamical systems and related stochastic processes, expected to be completed by June of this year. Chapter 1 is found here. Previous books in this series include “Stochastic Processes and Simulation”, “Intuitive Machine Learning and Explainable AI”, and “Synthetic Dara and Generative AI”. They are available here. Future books already started include “Statistical Optimization for Machine Learning” as well as “Experimental Math and Probabilistic Number Theory”.

In about 15 pages, this scratch course on chaotic systems covers a lot more material than expected in such a short presentation. Written in simple English yet covering topics ranging from introductory to advanced, it is aimed at practitioners interested in a quick, compact, easy-to-read summary on the subject. Students learning quantitative finance, physics or machine learning will also benefit from this material and the accompanying Python code. It is designed to help them understand concepts that are traditionally embedded in jargon and arcane theories. Numerous off-the-beaten-path exercises complement the presentation and may also be of interest to instructors and professors covering this topic: you are welcome to use them in your class! They require analytical and original thinking, in contrast to standard textbooks frequently featuring mechanical exercises.

Four basins of attraction of the 2D sine map, each with its own color

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.

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The technical article, entitled Introduction to Discrete Chaotic Dynamical Systems, is accessible in the “Free Books and Articles” section, here. It contains links to my GitHub files, to easily copy and paste the code. The text highlighted in orange in this PDF document are keywords that will be incorporated in the index, when I aggregate all my related articles into books about machine learning, visualization and Python. The text highlighted in blue corresponds to external clickable links, mostly references. And red is used for internal links, pointing to a section, bibliography entry, equation, and so on.

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About the Author

Vincent Granville is a pioneering data scientist and machine learning expert, co-founder of Data Science Central (acquired by  TechTarget in 2020), founder of, former VC-funded executive, author and patent owner. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Vincent is also a former post-doc at Cambridge University, and the National Institute of Statistical Sciences (NISS).  

Vincent published in Journal of Number TheoryJournal of the Royal Statistical Society (Series B), and IEEE Transactions on Pattern Analysis and Machine Intelligence. He is also the author of multiple books, including “Intuitive Machine Learning and Explainable AI”, available here. He lives  in Washington state, and enjoys doing research on spatial stochastic processes, chaotic dynamical systems, experimental math and probabilistic number theory.

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