Heteroclinic Orbits for Retarded Functional Differential Equations

Abstract Suppose Γ is a heteroclinic orbit of a Ck functional differential equation x (t) = ƒ(x i ) with α-limit set α(Γ) and ω-limit set ω(Γ) being either hyperbolic equilibrium points or periodic orbits. Necessary and sufficient conditions are given for the existence of an ftf close to ƒ in Ck with the property that x (t) = \ tf(x t ) has a heteroclinic orbit \ gG close to Γ. The orbits \ gG are obtained from the zeros of a finite number of bifurcation functions G(β, f )∈R d ∗ ,β ∈R d + 1 . Transversality of Γ is characterized by the nondegeneracy of the derivative DβG. It is also shown that the \ tf which have heteroclinic orbits near Γ are on a Ck submanifold of finite codimension = max{0, − indΓ} or on the closure of it, where ind Γ is the index of Γ.