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.
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$\begingroup$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
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- 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.
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