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If we have a Markov coding or another symbolic description of a dynamical system it is usually easy to prove that the system is chaotic (in the sense of of Li-York, Devaney, positive entropy of what ever). My impression is that most (interesting) dynamical systems coming of natural science are in fact chaotic. But this is rigourously proved only for a few systems. What is the reason for this? Is it difficult to find a symbolic coding for systems coming from natural science or are we (as mathematicians) not so much interested in these systems?

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Proving that a mathematical model is (or isn't) chaotic is not quite the same thing as proving that Nature is (or isn't) chaotic. –  Gerry Myerson May 8 '13 at 6:09
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The proofs of chaotic behaviour tend to rely on hyperbolic behaviour (or at least non-uniformly hyperbolic behaviour). Proving that this holds in many real systems (or even in lots of toy models) is extremely hard / apparently beyond the reach of current technology. –  Anthony Quas May 8 '13 at 8:26
    
Dear Gerry, of course You are right. Please apologize the provoking title. –  Jörg Neunhäuserer May 8 '13 at 16:30
    
Dear Anthony, i understand what You mean. But sometimes I wonder if it is necessary to use the machinery of (non-uniform) hyperbolic dynamics to prove that a systems is chaotic. If you directly construct an appropriate dynamical partition, you may prove that a system is chaotic without showing hyperbolicity (or?). In fact beside toy models I do not know a strategy to find such a partition, but may be there is a (black magic) way. –  Jörg Neunhäuserer May 8 '13 at 16:42
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