Normal forms of piecewise-smooth monodromic systems

Abstract Normal form is an important tool in the study of bifurcations but, unlike those for smooth differential systems, normal forms for piecewise-smooth systems have difficulty with the near-identity transformation, which is constructed piecewise but needs to be homeomorphic. Such a difficulty of homeomorphism was encountered when we simultaneously normalized two matrices and obtained the second-order normal form near an equilibrium of FF type for piecewise-smooth monodromic systems. In this article, we establish normal forms for piecewise-smooth monodromic systems near an equilibrium of FF, FP, or PP type. We overcome the difficulty of homeomorphism and generalize both from second-order to any given order and from FF type to all FF, FP, and PP types. This new method can be used to compute Lyapunov constants and discuss degenerate Hopf bifurcations for piecewise-smooth systems.