There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.
Tails of exit times from unstable equilibria on the line