Cyclic hybrid systems of flows on manifolds

The considered continuous-and-discrete hybrid system is a cyclic relay of smooth flows on an $n$-dimensional manifold $M$, where the discrete process of switching from each flow to the next takes place on the boundaries of the corresponding $n$-dimensional submanifolds of $M$. The main result is the existence of at least one periodic cyclic trajectory under a certain topological condition concerning one of the submanifold boundaries.