★★★★★ Extraordinarily lucid exposition of an extraordinarily difficult subject

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering

Steven H Strogatz

My review of the first edition of Nonlinear Dynamics and Chaos follows. Everything I say there applies to the second edition as well. Aside from minor corrections, the second edition differs from the first mainly in the addition of new exercises. Strogatz writes in the Preface

In the twenty years since this book first appeared, the ideas and techniques of nonlinear dynamics and chaos have found application in such exciting new fields as systems biology, evolutionary game theory, and sociophysics. To give you a taste of these recent developments, I’ve added about twenty substantial new exercises that I hope will entice you to learn more.

Review of the First edition

Nonlinear Dynamics is one of the most difficult areas of Applied Mathematics, but you would hardly guess that from reading Steven H. Strogatz. You can read Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering like a (very expensive -- but your university library surely has a copy) novel: start at the beginning, and read one page after another until you get to the end. It starts out simple and builds -- read in order, it all makes sense -- especially if you do the exercises. (Now, be clear: this is a math book. If math is not your thing, you're not likely to enjoy it.) Few mathematics books, even those about far more tractable subjects, are this readable.

What is "nonlinear dynamics"? "Dynamics" means, roughly, things that change with time. For instance, a car driving from your home to the university library is a dynamic system, and so is a field full of crops that are planted and grow, and so is a disease spreading through a population. This is obviously a big and important subject, and mathematicians have spent much effort on it over the years. They have had most success with linear dynamic systems.

I'm not going to give a technical definition of "linear" -- if you're a mathematician you already know -- but in practice it means you can solve a complicated linear problem by breaking it down into simple subproblems whose solutions are known, then combining those simple solutions to produce the solution of the complicated thing. It is the "combining" that linearity makes simple. Linear dynamics is boring (1) because it is mostly a solved problem, and (2) because certain really cool things can never happen in a linear system. For instance, you can have an explosion in a linear system, but there is no way for the explosion to end. If you want to describe a world in which explosions happen, and then stop, and then happen again, you need nonlinearity.

Almost nothing in The Real World™ is truly linear. However, many, many things in The Real World™ are approximately linear. (This is, more or less, what calculus is all about.) Thus linear dynamics allows us to describe all sort of things just up to the point where they get Really Interesting And Difficult. Strogatz is here to tell you about the "Really Interesting" stuff.

One thing I like about Strogatz's style is that he works hard to make things clear. There are lots of pictures. Also, he is free of the Pointless Purity fetish that afflicts so many mathematicians. He says, in so many words

Throughout this chapter we have used graphical and analytical methods to analyze first-order systems. Every budding dynamicist should master a third tool: numerical methods.

That is, you are allowed, indeed encouraged, to use a computer! You can never (well seldom) rigorously prove anything with numerical methods, and since proofs are what mathematics is all about, some mathematicians scorn numerical methods. Now, Strogatz is a mathematician. He knows what rigor is and employs it when it's the best way to an answer. But proofs in nonlinear dynamics are difficult, numerical methods are comparatively easy, and he uses both.

This is the best introduction to Nonlinear Dynamics in existence. If you have any interest in the subject, you should read it. Even if you think you have no interest in the subject, it's worth a look -- you might discover a new love.

Amazon review of the second edition

Goodreads review of the second edition

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