Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems

A topological approach and understanding to the detection of unstable perio dic orbits based on a recently proposed method [Phys. Rev. Lett. 78, 4733 ( 1997)] is developed. This approach provides a classification of the set of transformations necessary for finding the orbits. Applications to the Ikeda and Henon map are performed, allowing a study of the distributions of Lyap unov exponents for high periods. In particular, the properties of the least unstable orbits up to period 36 are investigated and discussed.