Stochastic dynamics on manifolds based on novel geometry preserving Ito–Taylor scheme

Abstract

This Rapid Communication presents the development of a new Ito–Taylor based higher order geometric numerical integration scheme for stochastically excited systems on manifolds. For the dynamical systems evolving on manifolds, two key motivations for the proposed work are: geometry preservation and accurate solution, which are not well addressed in the current state of the art. The accuracy of the proposed method as compared to the existing scheme (geometric Euler Maruyama) is achieved by incorporating the Kolmogorov operators in the proposed formulation. Preservation of geometry is ensured by exploiting the correspondence between the manifold and the tangent space at identity (Lie algebra). The efficacy of the proposed algorithm is demonstrated for non-linear oscillators and pendulum cart system, which are canonical representations of various physical phenomena in modern engineering problems.