Fixed point index for iterations of maps, topological horseshoe and chaos

Abstract

There are many examples of complicated or chaotic dynamics, but the set of examples for which chaos has been rigorously demonstrated is quite small. In most cases where chaotic dynamics has been proven, the strategy has involved analysing a simple singular map or integrable problem and then perturbing the results (see [2], [5]). This usually required some estimates on the derivatives of mappings under consideration. Another strategy to tackle such problems is to appropriately homotope the given system to a model problem for which some algebraic invariants could be explicitly computed and show that these invariants remain unchanged. Nontriviality of the algebraic invariant provides a minimal description of the complexity of the dynamics of the system. In [3], [4] with the help of the discrete Conley index introduced in [6], this strategy has been applied to the Henon map and the Lorenz equations. In applying this strategy to a concrete problem we must answer three closely related questions: what algebraic invariants we will use, what is the model map, what are the appropriate homotopies.