New Book: Gentle Introduction To Chaotic Dynamical Systems

In less than 100 pages, the book covers all important topics about discrete chaotic dynamical systems and related time series and stochastic processes, ranging from introductory to advanced, in one and two dimensions. State-of-the art methods and new results are presented in simple English. Yet, some mathematical proofs appear for the first time in this book: for instance, about the full autocorrelation function of the logistic map, the absence of cross-correlation between digit sequences in a family of irrational numbers, and a very fast algorithm to compute the digits of quadratic irrationals. These are not just new important if not seminal theoretical developments: it leads to better algorithms in random number generation (PRNG), benefiting applications such as data synthetization, security, or heavy simulations. In particular, you will find an implementation of a very fast, simple PRNG based on millions of digits of millions of quadratic irrationals, producing strongly random sequences superior in many respects to those available on the market.

An unusual orbit

Without using measure theory, the invariant distributions of many systems are discussed in details, with numerous closed-form expressions for classic and new maps, including the logistic, square root logistic, nested radicals, generalized continued fractions (the Gauss map), the ten-fold and dyadic maps, and more. The concept of bad seed, rarely discussed in the literature, is explored in details. It leads to singular fractal distributions with no probability density function, and sets similar to the Cantor set. Rather than avoiding these monsters, you will be able to leverage them as competitive tools for modeling purposes, since many evolutionary processes in economy, fintech, physics, population growth and so on, do not always behave nicely.

One orbit of the 2D sine map

A summary table of numeration systems serves as a useful, quick reference on the subject. Equivalence between different maps is also discussed. In a nutshell, this book is dedicated to the study of two numbers: zero and one, with a wealth of applications and results attached to them, as well as some of the toughest mathematical conjectures. It will appeal in particular to busy practitioners in fintech, security, defense, operations research, engineering, computer science, machine learning, AI, as well as consultants and professional mathematicians. For students complaining about how hard this topic is, and deterred by the amount of advanced mathematics, this book will help them get jump-started. While the mathematical level remains high in some sections, it is explained as simply as possible, focusing on what is needed for the applications.

Fractal approximation error linked to singular distributions

Numerous illustrations including beautiful representations of these systems (generative art), a lot of well documented Python code, and nearly 20 off-the-beaten path exercises complementing the theory, will help you navigate through this beautiful field. You will see how even the most basic systems offer such an incredible variety of configurations depending on a few parameters, allowing you to model a very large array of phenomena. Finally, the first chapter also covers time-continuous processes including unusual clustered, reflective, constrained, and integrated Brownian-like processes, random walks and time series, with little math and no obscure jargon. In the end, my goal is to get you to you use these systems fluently, and see them as gentle, controllable chaos. In short, what real life should be! Quantifying the amount of chaos is also one of the topics discussed in the book.

Authored by Dr. Vincent Granville, 82 pages, published in March 2023. Available on our e-Store exclusively, here. See the table contents or sample chapter on GitHub here. The Python code is also in the same repository.

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