Existence of periodic orbits for high-dimensional autonomous systems

We give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension. It is based on a Poincaré–Bendixson property enjoyed by a new class of monotone systems introduced in [L.A. Sanchez, Cones of rank 2 and the Poincaré–Bendixson property for a new class of monotone systems, J. Differential Equations 216 (2009) 1170–1190]. A concrete application is done to a scalar differential equation of order 4.