  Poincar\'e maps are an integral aspect to our understanding and analysis of
nonlinear dynamical systems. Despite this fact, the construction of these maps
remains elusive and is primarily left to simple motivating examples. In this
manuscript we propose a method of data-driven discovery of Poincar\'e maps
based upon sparse regression techniques, specifically the sparse identification
of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the
dynamics on and near invariant manifolds of a given dynamical system, as well
as provide long-time forecasting of the coarse-grained dynamics of multiscale
systems. Moreover, the method provides a mathematical formalism for determining
nonlinear Floquet theory for the stability of nonlinear periodic orbits. The
methods are applied to a range of examples including both ordinary and partial
differential equations that exhibit periodic, quasi-periodic, and chaotic
behavior.
