Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center

In this paper we prove that any Σ-center (either nondegenerate or degenerate) of a planar piecewise CrCr smooth vector field ZZ is topologically equivalent to that of Z0Z0: (x˙,y˙)=(−1,2x) for y≥0y≥0, (x˙,y˙)=(1,2x) for y≤0y≤0, and that the homeomorphism between ZZ and Z0Z0 is CrCr smoothness when restricted to each side of the switching line except at the center p.We illustrate by examples that there are degenerate Σ-centers whose flows are conjugate to that of Z0Z0, and also there exist nondegenerate Σ-centers whose flows cannot be conjugate to that of Z0Z0.Finally applying the normal form Z0Z0 together with the piecewise smooth equivalence, we study the number of limit cycles which can be bifurcated from the Σ-center of ZZ.