What defines a chaotic system?

Question

Suppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My question is, how do we actually prove this? Also, is it necessary for a chaotic system to be chaotic in any coordinate system, or do we have to prove it for every coordinate system?

I'd guess that if a system is chaotic for one set of coordinates, it'll be chaotic for almost every other coordinate system - except for when you specifically choose to fix the equation.

Answer 1

There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system

1. has sensitive dependence on initial conditions
2. is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
3. its periodic orbits form a dense set

It was found later that 1. is a consequence of 2. and 3.

Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.