Approximation of Vector Fields on the RG Method and its Application to the Synchronization

The renormalization group (RG) method for differential equations is one of the perturbation technique proposed by Chen, Goldenfeld, and Oono [1,2]. The RG method unifies traditional singular perturbation methods, such as the multi-scaling method, the boundary layer theory , the averaging method, the normal form theory, the center manifold theory, and the geometric singular perturbation. Chiba [3] proved that a family of approximate solutions constructed by the RG method defines a vector field which is approximate to an original vector field (ODE) in C1 topology. By using this result, we can show that if the RG equation has a normally hyperbolic invariant manifolds N, the original equation also has an invariant manifold which is diffeomorphic to N [3].