# Definition of chaos for maps

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.

## 2 Answers

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).

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