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.

On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

Fast phase randomization via two-folds

A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and, in the presence of small noise, the asymptotic phase of a large number of sample solutions is highly random. In this paper, we show how the probability distribution of the asymptotic phase depends on the global nonlinear dynamics. We also show how the phase of a smooth oscillator can be randomized by applying a simple discontinuous control law that generates an invisible two-fold. We propose that such a control law can be used to desynchronize a collection of oscillators, and that this manner of phase randomization is fast compared with existing methods (which use fixed points as phase singularities), because there is no slowing of the dynamics near a two-fold.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.