91Ó°ÊÓ

Can an autonomous vector field on \(\mathbb{R}\) that has no equilibrium points have periodic orbits? We assume that existence and uniqueness of solutions holds.(You must justify your answer.) \(^{3}\)

Show that the time shift property for autonomous vector fields implies that trajectories cannot "cross each other", i.e. intersect, in phase space.

Show that the intersection of two invariant sets is an invariant set.