Bifurcation from a Heteroclinic Solution in Differential Delay Equations

We study a class of functional differential equations x(t) = af(x(t 1)) with periodic nonlinearity /: R -> R, 0 < / in ( A, 0) and / < 0 in (0, B ), /( A ) = /(0) = /( B) = 0. Such equations describe a state variable on a circle with one attractive rest point (given by the argument £ = 0 of/) and with reaction lag a to deviations. We prove that for a certain critical value a = a0 there exists a heteroclinic solution going from the equilibrium solution / -» A to the equilibrium / -> B. For a a0 > 0, this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle. Introduction. Consider a state variable on a circle with one attractive rest point, and with a delayed reaction to deviations. A simple differential equation for such a system is y(s)=f(y(s-a)), a>0, or equivalently (af) x(t) = af{x{t-l)), where the function /: R -> R is periodic with minimal period -A + B, A < 0 < B, f(A) = 0 = f(0) =f(B),0 0. This corresponds to a periodic rotation of the state variable on the circle. Such solutions cannot arise in local bifurcation from equilibria. In the theory of O.D.E.'s, say for vector fields on a cylinder, creation and destruction of periodic solutions which wind around the cylinder are well known. Received by the editors July 24, 1984. 1980 Mathematics Subject Classification. Primary 34K15.