Where can I learn "everything" about strange attractors?

Question

I have a (possibly very) limited math background, but I would like to find a book, website, or other type of work that will teach almost everything about strange attractors.

As a motivating example, I found a recurrence of the following form:

$$x_{n+1} = a x_n + y_n$$ $$y_{n+1} = b + (x_n)^2$$

...at this Mathematica website. I'm very interested in finding more exact values (for $a$ and $b$) for the recurrence, and knowing how to do this. I'd also like to be able to prove that this defines a strange attractor. So anything that would allow me to do this would be great, but I'd be especially interested in finding a "Bible" on chaos and strange attractors, kind of like Artin's Algebra, Herbert Wilf's Generatingfunctionology, or some other work that can be considered to be the guide to the subject.

Answer 1

If you have a limited mathematical background, I think starting with the classic Nonlinear Dynamics & Chaos by Steven Strogatz is a great start. It is not specifically about strange attractors, but it does give you the foundations for it (which I do strongly feel one needs), and it is very easy to read with a lot of applied examples from various fields, from physics to biology.

Answer 2

The book Chaos and Fractals - New Frontiers of Science here by Peitgen/Jürgens/Saupe contains a chapter devoted to strange attractors. There is for sure no book which contains "almost everything" but this is a good starting point. And now I explain why this would be a good starting point.

Strange attractors originate from dynamical systems, i.e. systems of ordinary differential equations. The recurrence you've posted might be a discretization of such a "chaotic" system using a simple numerical solver. First of all, the chapter about Strange Attractors in this book starts with a historical introduction, i.e. the first papers about this topic. Then an example is given followed by a definition of the term Strange Attractor. More example follows with very detailed analytic discussion of them. We also find the most popular one's: The Rössler and the Lorenz Attractor. What I like most on this book it is analytic rigour which might be hard if you do not have a very strong background on mathematics. However, since the part about Strange Attractors occur in the second half of the book, you should read through the other stuff. But this is possible, since the book is "almost" self-contained, except some standard mathematical notions like sequences and so on.

I cite from the books backcover:

Fascinating and authoritative, Chaos and Fractals: New Frontiers of Science is a truly remarkable book that documents recent discoveries in chaos theory with plenty of mathematical detail, but without alienating the general reader. In all, this text offers an extremely rich and engaging tour of this quite revolutionary branch of mathematical research...

and

This book - which has been named "Most outstanding book in the mathematics category" contains all one ever wanted to know about fractals, and more. Written by - next to Mandelbrot - the greatest popularizer of the concept of fractal geometry it contains a wealth of information on nearly every angle of the topic...

Answer 3

I don't go in for mathematical bibles as a general policy, so I'll make a different kind of recommendation.

In brief, learn the theory of Axiom A dynamical systems, and while you're doing that learn about symbolic dynamics and Markov partitions.

Look at Smale's 1967 article in the Bulletin of the American Mathematical Society entitled "Differentiable Dynamical Systems", and the little book in which Ruelle's lecture notes "Chaotic evolution and strange attractors" is published.

Read the mathematical papers of Rufus Bowen and of David Ruelle.

By then, you'll have learned a little tiny bit about strange attractors.