Fast chaos expansions of diffusive and sub-diffusive processes in orbital mechanics

This paper considers the case in which the dynamics of an object in orbit is subject to a random process that can be modelled as a generic Continuous Time Random-Walk (CTRW). This model can describe the situation in which an orbiting object is subject to random, non destructive, collisions. In the limit of an infinite number of collisions the process can converge to a standard Weiner process if the impacts occur at time increments that are stationary with finite variance. In this case it can be shown that the dynamics is subject to normal diffusion. However, impact can occur more slowly with time increments drawn from a heavy tailed distribution, in this case the dynamics is subject toa sub-diffusion process. The paper discusses how polynomial expansions with dynamic re-initialisation can be used to study the evolution of the dynamical system subject to a CTRW. A simple indicator of diffusion is then derived from the coefficients of the polynomial expansion. The same indicator is then used to study regular and chaotic motions in a dynamical system with uncertain model parameters. A few simple illustrative examples will complete the paper.