  We continue the approach in Part I \cite{duchong19} to study stationary
states of controlled differential equations driven by rough paths, using the
framework of random dynamical systems and random attractors. Part II deals with
driving paths of finite $\nu$ - H\"older norms with $\nu \in
(\frac{1}{3},\frac{1}{2})$ so that the integrals are interpreted in the
Gubinelli sense for controlled rough paths. We prove sufficient conditions for
the attractor to be a singleton, thus the pathwise convergence is in both
pullback and forward senses.
