On local bifurcations of second order differential equations with a piece-smooth right-hand side

The paper considers dynamical systems on the plane, given by second-order autonomous differential equations, with right-hand sides depending on one and two parameters, discontinuous on the “zero velocity line” y = 0 and smooth outside it. At zero values of the parameters, it is assumed that the origin of coordinates is a stable equilibrium "linked" from the foci of smooth dynamical systems specified in the halfplanes y 0. Bifurcation diagrams are described for generic one-parameter and two-parameter families of such systems. In particular, it is shown that in a one-parameter family, a single (stable) limit cycle can be born from the equilibrium, and in a two-parameter family, from one to two limit cycles can be born from an the equilibrium.