Existence of periodic orbits of autonomous retarded functional differential equations

Abstract

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.