Existence and Number of Figure-Eight Loops in Planar Sector-Wise Linear Systems with Saddle–Saddle Dynamics

In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.