Splitting of a system of differential equations with slowly varying phase in the neighborhood of an asymptotically stable invariant torus (Q1092305)

The authors consider the system (1) \(d\phi /dt=\epsilon a(\phi,y,\epsilon)\), \(dy/dt=P(\phi,y,\epsilon)\) where \(\phi =(\phi_ 1,...,\phi_ m)\), \(y=(y_ 1,...,y_ n)\) are respectively, cyclic and normal coordinates, and they relate the decoupling property of the system to existence of a function \(\Phi\) periodic in the cyclic coordinates and satisfying some boundedness and smoothness conditions. (The system (1) is decoupled if it can be rewritten in the form: \(d\theta /dt=\epsilon a_ 0(\theta,\epsilon)\), \(dy/dt=P(y).)\)