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 LiYork, 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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