Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type

The existence of a self-sustained periodic solution in the autonomous equation m'(t) — au' (t — h) + /3u(t) + ay u(t — h) = «/(w(r)) is proved under appropriate assumptions on a, ft, y, f and h. The method of proof consists in converting this equation into an equivalent nonlinear integral equation and demonstrating the convergence of an appropriate iteration scheme. In this paper we consider the equation m'(t) — au'(r — h) + /3w(r) + ayu(r — h) = ef(u(T)), (1) where h > 0 and a = aa(l + e). The existence of a periodic solution will be proved for small t > 0 under appropriate assumptions on the parameters a0, (3, y, and /. The left-hand side of this equation is a linear difference-differential operator of neutral type (for a definition see [1]). The existence of periodic solutions for functional-differential equations which include difference-differential equations of retarded type but not neutral type has been discussed by Krasovskii [2], Shimanov [3], [4], and Hale [5]. In all these cases, the equations are of the forced type where the right-hand side is a 2?r-periodic function of r. Equation (1), which we consider, is autonomous, and we look for a self-sustained oscillation. Difference-differential equations of the type (1) arise from electrical networks such as the one shown in Fig. 1. The equations of this network are Udi/dt) = -dv/dx, 0 < ^ < J (2) C{dv/dt) = —di/dx, E — v(0, t) — Ri(0, t) = 0, Ci(dv(l, t)/dt) = i{ 1, t) — g(v( 1, t)), where L, C are the specific inductance and capacitance in the transmission line. The question of the existence of a periodic solution of some unknown period T can be posed by giving the additional boundary conditions v(x, 0) = v(x, T), i(x, 0) = i(x, T). Thus, we have a boundary-value problem for a hyperbolic partial differential equation with boundary conditions given on the rectangle shown in Fig. 2. Of course, in general, a boundary-value problem for a hyperbolic equation is not well posed. The difference here is that the boundary t — T is free to be chosen. •Received November 5, 1965; revised manuscript received March 4, 1966. The results reported in this paper were obtained in the couse of research jointly sponsored by the Air Force Office of Scientific Research (Contract AF 49(638)-1139) and IBM. 216 It. K. BRAYTON [Vol. XXIV, No. 3