Doctor of Philosophy, The Ohio State University, 2004, Mathematics

In the present thesis, we study a particular 3-D map with a parameter ε>0, which has two fixed points. One fixed point has a 1-D unstable manifold, while the other has a 1-D stable manifold. The main result is that we prove the smallest distance between theupdate.cgi two manifolds is exponentially small in ε for small ε. We first prove in the limit of ε → 0+, bounded away from +∞ or -∞, both the stable and unstable manifolds asymptotes to a heteroclinic orbit for a differential equation. Then we show there exists a parameterization of the manifolds so that they differ exponentially in ε. By examining the inner region around the nearest complex singularity of the limiting solution, and using Borel analysis, we relate the constant multiplying the exponentially small term to the Stokes constant of the leading order inner equation.

Committee: Saleh Tanveer (Advisor); Yuan Lou (Other); FeiRan Tian (Other) Subjects: Mathematics