| We develop a rigorous formalism for a computational theory for finding various sets important to the dynamics of a map. We develop a novel approach to locating basin boundaries and arbitrary isolated invariant sets. Our comprehensive theory also covers a number of published results. Examples of these include periodic orbits, chain-recurrent sets, and maximal invariant sets. We begin by describing "upper-bound sets", that is, sets that contain the desired set. In our theory, each upper-bound set is a collection of grid boxes. We give conditions which guarantee that as precision and resolution increase, the upper-bound sets converge to (shrink down to) the desired set. We describe a measure of closeness of particular upper-bound sets to the sets they contain. We prove that our implementation of the theory for C 2 maps achieves, in the sense of the measure, the closest possible upper-bound sets of this type.; This algorithm can in principle be implemented using any reasonable interval arithmetic, but a specific efficient implementation is presented here, along with some results from this implementation. |
