Local contraction analysis of stochastic systems with limit cycles

A method is presented to obtain inner estimates of the region of transverse contraction (ROTC) which are invariant regions in which trajectories of a stochastic system converge to a stochastic limit cycle. Using the framework of Polynomial Chaos Expansions (PCE) the stochastic system is represented by a higher dimensional deterministic system. First, the connection between the stability of the periodic orbits of the stochastic system and the stability of the limit cycle of its PCE system is established. Then transverse contraction criteria, as well as invariance conditions, are formulated for the PCE system to certify an ROTC estimate for the PCE system. From this, and by leveraging the established stability connection, an ROTC estimate of the stochastic system is retrieved. Finally, an optimization program, based on matrix sum-of-squares verification techniques, to implement the contraction and invariance criteria is proposed.