Center boundaries for planar piecewise-smooth differential equations with two zones

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary  . We prove that given a pair of systems that share a hyperbolic focus singularity p0p0, with the same orientation and opposite stability, and a ray Σ0Σ0 with endpoint at the singularity p0p0, we can find a smooth manifold Ω such that Σ0∪{p0}∪ΩΣ0∪{p0}∪Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.