  In this paper, we claim the availability of deterministic noises for
stabilization of the origins of dynamical systems, provided that the noises
have unbounded variations. To achieve the result, we first consider the system
representations based on rough path analysis; then, we provide the notion of
asymptotic stability in roughness to analyze the stability for the systems. In
the procedure, we also confirm that the system representations include
stochastic differential equations; we also found that asymptotic stability in
roughness is the same property as uniform almost sure asymptotic stability
provided by Bardi and Cesaroni. After the discussion, we confirm that there is
a case that deterministic noises are capable of making the origin become
asymptotically stable in roughness while stochastic noises do not achieve the
same stabilization results.
