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Our starting point is a discontinuous piecewise linear dynamical system in ℝ 3 with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a T -singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation. [ABSTRACT FROM AUTHOR]

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 ℒ invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly k 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 k ⋅ ℒ limit cycles. [ABSTRACT FROM AUTHOR]

We study the limit cycles of two families of piecewise-linear differential systems in R 3 with two pieces separated by a plane Σ . In one family the differential systems are continuous on the plane Σ , and in the other family they are discontinuous on the plane Σ . The usual tool for studying these limit cycles is the Poincaré map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been done with the algebraic manipulator Mathematica. [ABSTRACT FROM AUTHOR]

We consider a class of three-dimensional piecewise linear differential systems with two zones separated by a plane. In this class we show that there exist differential systems having: (i) a unique limit cycle; (i i) a unique one–parameter family of periodic orbits; (i i i) scrolls; (i v) invariant cylinders. • Characterization of limit cycles for 3D-sewing piecewise linear differential system. • Conditions for the existence of invariant surfaces: scrolls, cylinders, and cones. • Existence of a Hopf-Scroll: one limit cycle contained in a unique invariant cylinder. • Correspondence of existence of limit cycles with intersections of algebraic curves. [ABSTRACT FROM AUTHOR]