Cyclic differential equations and period three solutions of differential-delay equations

Abstract

where g: C?’ --) )? is a given function, Ni are integers and a is a real parameter. Such equations have been considered in several papers, for instance [ 2, 5, 6). It has become apparent that it is important to be able to show that (0.1) has no nontrivial periodic solutions of certain periods. For example, the global Hopf bifurcation theorem, as developed by Alexander and Yorke for ordinary differential equations and later extended to functional differential equations 12, 6 1, gives relatively little information about (0. I ) if one does not know that (0.1) has no nontrivial solutions of certain periods. A well-known result (see Lemma 4.1 in (21) asserts that Eq. (0. I ) can have no nonconstant periodic solutions of period 2. In this paper we show that for certain classes of g, (0.1) can have no nonconstant periodic solutions of period 3. The general idea that periodic solutions of (0.1) of period n, n an integer, satisfy an associated system of ordinary differential equations has been used before: see 141 and Lemma 4.1 of 121. However, in our case we deal with a system of three first order ODES, instead of two ODES as in Lemma4.1 of 121.