Melnikov theory for a class of planar hybrid systems

In this paper, we extend the well-known Melnikov theory for smooth systems to a class of two-zonal planar hybrid piecewise smooth systems. In this class, each zone of differentiability is separated by a straight line and in these zones, the dynamics are governed by a smooth system. When an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone. The family of non-generic systems of this class which has a continuum of periodic orbits is considered. Then, we study the periodic orbits of the continuum that persist under a perturbation. To achieve this objective, first we build an appropriate Poincaré map, after that, we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation. This function will be called the Melnikov function because it is very close to the so-called Melnikov function for smooth systems. Finally, we apply the obtained results to some piecewise linear systems, continuous as well as discontinuous ones.