Bifurcations from a center at infinity in 3D piecewise linear systems with two zones

Abstract

We consider continuous piecewise linear systems in R3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity. A specific conic projection is introduced, so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations.In the semi-homogeneous case, such a center is associated to the existence of a continuum of invariant semi-cones sharing the vertex at the origin; perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi-cones.For the non-homogeneous case, the cycles at infinity does not imply invariant semi-cones. However, the non-generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields, becomes rather interesting. It is possible then, by perturbing the other vector field, to get the bifurcation of a big limit cycle from infinity without destroying the center.