  Since the breakthrough in rough paths theory for stochastic ordinary
differential equations (SDEs), there has been a strong interest in
investigating the rough differential equation (RDE) approach and its numerous
applications. Rough path techniques can stay closer to deterministic analytical
methods and have the potential to transfer many pathwise ordinary differential
equation (ODE) techniques more directly to a stochastic setting. However, there
are few works that analyze dynamical properties of RDEs and connect the rough
path / regularity structures, ODE and random dynamical systems approaches. Here
we contribute to this aspect and analyze invariant manifolds for RDEs. By means
of a suitably discretized Lyapunov-Perron-type method we prove the existence
and regularity of local center manifolds for such systems. Our method directly
works with the RDE and we exploit rough paths estimates to obtain the relevant
contraction properties of the Lyapunov-Perron map.
