Reddit reviews Chaos: An Introduction to Dynamical Systems (Textbooks in Mathematical Sciences)

The iterative map x = cos(x) converges to the fixed point, the value of x for which x = cos(x). I don't think there's a nice analytic expression.

What's more interesting is why the map converges for all initial conditions. Firstly, the map x = cos(x) is what's called a dynamical system. An orbit for this dynamical system with initial condition x is the sequence {x, cos(x), cos(cos(x)), ...}. As you've found out by experimentation, all orbits converge to a single value. This value is called an attractor for the dynamical system, and an attracting fixed point in this case. To see why this is true, you can draw a cobweb plot. Wikipedia even has an illustration for the fixed point of x = cos(x).

I find the field of dynamical systems extremely interesting. In the Wikipedia article of the cobweb plot, there is an illustration of much more interesting behavior called chaos. A book that I can highly recommend, that starts with this kind of iterative maps and works its way up is Chaos: An Introduction to Dynamical Systems by Alligood, Sauer and Yorke, though it requires knowledge of Calculus.

For the most part, a complex system is just a dynamical system that isn't well understood. Choas theory is a small part of dynamical systems.

You only need up to an undergraduate differential equations class to understand chaos theory. A good book is Alligood and Yorke

As far as textbooks go, Alligood's Choas is a great introduction to Dynamical Systems. It might be a little advanced (it's designed for late undergrad or early grad courses), but I find it to be very accessible, and it includes a lot of supplementary material on the topic (plus pretty pictures).

Also, one of the authors (James Yorke) is a pretty big player in Chaos Theory; he coined many terms such as "chaos", "crisis", etc. in the landmark paper "Period 3 Implies Chaos" (which is actually a document included in the first chapter of the book, and is pretty accessible).

Chaos: An Introduction to Dynamical Systems by Alligood et. al. It can easily be used for self study. The book covers some elementary topics almost to the point of exhaustion, but in doing so unveils some really useful general techniques. Some of the exercises and labs will require some computer programming skills; Mathematica or the pythonxy environment will do everything you need and a billion things more.