An Algorithm for the Computation of Preimages in Noninvertible Mappings

Abstract

For discrete dynamical systems generated by iterating a diffeomorphism, every point in the phase space has a unique preimage and it is straightforward to compute geometric structures such as inverse orbits and one-dimensional stable manifolds of periodic points. For noninvertible mappings, however, some points have multiple preimages; others may have no preimages. This makes the computation of inverse orbits difficult, because accurate computations require global knowledge about the way the mapping folds and pleats phase space. In this article we use ideas from singularity theory to examine the geometry of noninvertible mappings. We use the geometry to derive a computational algorithm for efficiently computing preimages in noninvertible mappings.