The Poincaré Map

Abstract

We begin by recalling the definition of the Poincaré map. In doing this, one has to prove that the mapping exists and is differentiable. In fact, one can do this in the context of C⁰ flows F_t(x) such that for each t, F_t is C^k, as was the case for the center manifold theorem for flows, but here with the additional assumption that F_t(x) is smooth in t as well for t > 0. Again, this is the appropriate hypothesis needed so that the results will be applicable to partial differential equations. However, let us stick with the ordinary differential equation case where F_t is the flow of a C^k vector field X at first.