I. INTRODUCTION (digest)

In this paper we continue earlier studies 1-3 of periodic and chaotic attractors observed in numerical simulations of the delay-differential equation:

 , (1)

where the time delay L and the right hand side δ are constants. This equation is a mathematical model of a phase-locked loop (PLL) with time delay.

Equation (1) can be rewritten in a different form by integrating from an initial time t0 to t:

 . (2)

In this integral form it is clear that the system acts on initial conditions in the space of integrable functions on some time interval of length L. Since the system (1) is autonomous, the choice of initial time is arbitrary, and we take t0 = 0, so that an initial function is given on [-L, 0]. If the initial function is continuous, a unique solution exists for t > 0 and the solution has a continuous derivative for t > 0; there may be a discontinuity in the derivative of θ at t = 0 (and in higher derivatives at t = L, 2L, ...) unless the initial function satisfies a compatibility condition.

It will be convenient to represent solutions of Eq. (1) or (2) as trajectories projected onto the phase plane with coordinates (θ, /dt). Since the full phase space of the system is an infinite-dimensional function space, we must expect that trajectories projected onto the phase plane may in some instances appear to cross themselves. (In fact for almost all the steady states to be considered here, it seems that the apparent self-crossings could be removed by adding a third coordinate, say d2θ/dt2, to the projection; but see Ref. 3 for phase portraits which suggest structure in more than three dimensions.) In addition to the projected phase plane with -∞ < θ < ∞, it is also useful to consider the cylindrical identification space in which each θ is identified with θ + 2 for each integer n, and our phase portraits will be presented in a rectangular region [-π, π) X [δ - 1, δ + 1] unwrapped from the cylindrical identification space. (It is clear from Eq. (1) that /dt will never fall outside [δ - 1, δ + 1].)