3-dimensional piecewise linear and quadratic vector fields with invariant spheres

Abstract

We consider the class $\mathcal{X}$ of $3$-dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence of at least ten $1$-parameter families of crossing closed trajectories for quadratic vector fields in $\mathcal{X}$.

Keywords

• piecewise smooth vector fields with invariant spheres
• invariant cones
• $1$-parameter families of closed trajectories