Condition of Chaos: Ljapunov Exponent & Strange Attractor

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A positive Lyapunov exponent is commonly associated with chaotic behavior, while strange attractors are indicative of such dynamics. The discussion raises the question of whether there are theorems that establish the equivalence of these conditions as necessary and sufficient for chaos. It highlights that topological mixing and sensitive dependence on initial conditions are necessary for chaos. Additionally, there is a request for resources that provide a precise statement and proof regarding the necessity of non-invertibility for chaos in one-dimensional systems. Understanding these relationships is crucial for deeper insights into chaotic systems.
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Usually positive Ljapunov exponent is said to be a condition for chaos,
and the strange atractor is so.

Are there any theorem which shows that the two condition is equivalent?
They are necessary and sufficent conditions?
 
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The necessary conditions for chaos are topological mixing and sensitive dependence of initial conditions.
 
Where can I find a precise statement and proof of "non-invertibility is necessary condition for chaos in 1D" ?
 

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