Emulation of Poincaré return maps with Gaussian Kriging models

In this paper we investigate the use of Gaussian emulators to give an accurate and computationally fast method to approximate return maps, a tool used to study the dynamics of differential equations. One advantage of emulators over other approximation techniques is that they encode deterministic data exactly, so where values of the return map are known these are also outputs of the emulator output, another is that emulators allow us to simultaneously emulate a parameterized family of ODEs giving a tool to assess the behavior of perturbed systems. The methods introduced here are illustrated using two well-known dynamical systems: The Rossler equations, and the Billiard system. We show that the method can be used to look at return maps and discuss the further implications for full computation of differential equation outputs