Invariant manifolds of 3D piecewise vector fields

We analyze a 3D piecewise linear dynamical system $Z=(X,Y)$ with a plane Σ as its switching manifold containing two-fold parallel straight lines. The eigenvalues associated with X and Y are composed of two complex eigenvalues and one non-zero real eigenvalue. Using a suitable canonical form and exponential matrices theory, we generate two closing equations, from which we derive two half-return Poincaré maps. By defining the displacement map as the difference between the two half-return Poincaré maps from the same point, we prove using the Weierstrass preparation theorem that there exists a 3D piecewise linear dynamical system that admits three invariant cylinders of big amplitude, with exactly one limit cycle in each cylinder, a surface cone-like cylinder, and a cylinder filled with closed orbits. Lastly, we provide examples of 3D piecewise linear dynamical systems that present three limit cycles, a cone-like surface, and a cylinder filled with closed orbits, respectively.