In my dissertation I have studied a discrete dynamic system given by a map, called the Smale Horseshoe map. The latter is determined by a simple rule, which tranforms a rectangle into a subtler shape and admits an extremely chaotic dynamic behaviour. More precisely, it turns out that this system is chaotic already when we restrict our attention to a part of orbits, which begin in a Cantor set of the initial rectangle and can be described with a space of sequences (symbolic dynamics). I used Smale's Horseshoe map also to explore an appearance of a homoclinic tangle - transversal intersection of the stable and unstable manifold.
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