Definition of chaos for maps

Question

I'm looking for a mathematical precise definition of chaos for maps as I don't find that much information on it on the web.

Answer 1

I studied at least two ways of defining chaos on real-valued maps during my Chaos class. The first looks at the orbit. (Chaos: An Introduction to Dynamical Systems - Authors: Alligood, Kathleen T., Sauer, Tim, Yorke, James).

CHAOTIC ORBITS

Let $f$ be a map of the real line R andlet {$x_1,x_2,...$} be a bounded orbit of $f$. The orbit is chaotic if:

1) {$x_1,x_2,...$} is not assymptotically periodic;

2) the Lyapunov expoent $h(x_1)$ is greater than zero.

CHAOTIC MAPS

The other way of defining chaos for maps is looking at the map itself. Here we have an invariant set $I$ such that $f:I\to I$.

We say the map is chaotic if

1) periodic points are dense in $I$;

2) there is an dense orbit in $I$;

3) there is sensitive dependense to initial conditions.

Check out Yorke's book for more details and definitions used here you don't get (for example Lyapunov expoent).

Answer 2

The book Chaotic Dynamics on Nonlinear Systems by S. Neil Rasband is an accessible introduction to the theory,techniques and applications of chaos.