Analysis of stochastic nearly-integrable dynamical systems using polynomial chaos expansions

In this paper we propose the use of dynamic intrusive Polynomial Chaos Expansions (dPCE) to study some properties of nearly-integrable systems in orbital mechanics, where the perturbation is stochastic; we focus on random-walk type of perturbations.

We use a simple Weiner process to model the stochastic component of the perturbation and a truncated Karhunen–Lo`eve expansion of the Weiner process to allow the treatment with Polynomial Chaos. In particular, we use dynamic Polynomial Chaos, where the integration time is divided in segments and PCEs are restarted on each segment, to keep the number of coefficients of the Karhunen–Lo`eve expansion contained.

We first study a stochastic version of the H´enon-Heiles system, we then consider the motion of a stochastically perturbed satellite in geostationary orbit. For both problems we show evidence of diffusion induced by the stochastic perturbation.