Taylor-like approximations of center manifolds for rough differential equations

Abstract: The dynamics of rough differential equations (RDEs) has recently received a lot of interest. For example, the existence of local random center manifolds for RDEs has been established. In this work, we present an approximation for local random center manifolds for RDEs driven by geometric rough paths. To this aim, we combine tools from rough path and deterministic center manifold theory to derive Taylor-like approximations of local random center manifolds. The coefficients of this approximation are stationary solutions of RDEs driven by the same geometric rough path as the original equation. We illustrate our approach for stochastic differential equations (SDEs) with linear and nonlinear multiplicative noise.