Optimization and the miranda approach in detecting horseshoe-type chaos by computer

We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form T ( W). U, and thus check certain suffcient conditions for chaotic behavior of dynamical systems in a rigorous way. Our proof of the underlying abstract theorem avoids referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem. The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach. Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Henon mapping.