Exact Numerical Methods in Dynamical Systems Theory

Interval mathematics is applied to computational problems in dynamical systems theory. We present methods of global analysis that automatically verify the existence and uniqueness of all periodic orbits of fixed period of iterated mappings. Furthermore, the method is generalised to include parameter space in the global analysis. A method is developed that computes a guaranteed enclosure of invariant sets of iterated mappings; the invariant sets may contain chaotic attractors or chaotic repellers. The methods are applied to the logistic map, the Hénon map and a two-dimensional trigonometric map.