What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos?
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The dimension should be 3 or larger. If the dimension is smaller then 3 the existence and uniqueness theorem for differential equations will tell you that functions can't intersect (since you want them to be continuous and differentiable).
In 1 dimension this means you can only have movement in one direction
In 2 dimensions this means that your value either goes to infinity or to a perticular point.
You need 3 dimensions or more to get these strange attractors and that weird chaotic behaviour.
According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are:
Deterministic system (only one future for each state)
Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature)
A positive maximum Lyapunov exponent.
3) is pretty much the quantitative standard in journals of chaos, assuming you meet the conditions of 1). 2) is subjective and there's things like "stable chaos" and there can be periodic behavior that appears irregular but just has a really long period before it repeats itself, so you have to be careful with 2).
