Limit Cycles on Piecewise Smooth Vector Fields with Coupled Rigid Centers

We provide an upper bound for the maximum number of limit cycles of the class of discontinuous piecewise differential systems formed by two differential systems separated by a straight line presenting rigid centers. These two rigid centers are polynomial differential systems with a linear part and a nonlinear homogeneous part. We study the maximum number of limit cycles that such a class of piecewise differential systems can exhibit.

Birth of Isolated Nested Cylinders and Limit Cycles in 3D Piecewise Smooth Vector Fields with Symmetry

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving rise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First, we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly [Formula: see text] invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly [Formula: see text] isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly [Formula: see text] limit cycles.

Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems

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.