More on Ordinary Differential Equations Which Yield Periodic Solutions of Delay Differential Equations

We construct a Poincaré operator for the system dxdt = −λx − F(x), (0.1) where λ is a real parameter, x ∈ R3, x = (x1, x2, x3), [formula], and ƒ is an odd C2 function such that ƒ′(0) = 1, xƒ(x) > 0, for x ≠ 0. We also consider the case where ƒ is C1. We will express F in linearized form, that is, F(x) = Ax + G(x), where A is the linearized part of F around zero and G(x) = o(|x|) near zero. Fixed points of the Poincaré operator correspond to periodic solutions of the functional differential equation dxdt = −λx(t) − ƒ(x(t − T/3)), (0.2) where T is the period of x.