3-dimensional piecewise linear and quadratic vector fields with invariant spheres - Buzzi, Claudio A. [UNESP] (2024)
We consider the class 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 X.