Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

Abstract

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).