A system of differential equations (DDEs) with constant delays has the following form:
Here, t is the independent variable, y is a column vector of dependent variables, and y ′ represents the first derivative of y with respect to t. The delays, τ1,…,τk, are positive constants.
The solutions of DDEs are generally continuous, but they have
discontinuities in their derivatives. The
tracks discontinuities in low-order derivatives. It integrates the
differential equations with the same explicit Runge-Kutta (2,3) pair
and interpolant used by
ode23. The Runge-Kutta
formulas are implicit for step sizes bigger than the delays. When y(t)
is smooth enough to justify steps this big, the implicit formulas
are evaluated by a predictor-corrector iteration.
Equation 11-1 is a special case of
that involves delays, dy1,..., dyk, which can depend on both time, t, and state, y. The delays, dyj(t, y), must satisfy dyj(t, y) ≤ t on the interval [t0, tf] with t0 < tf.
finds the solution, y(t), for
DDEs of the form given by Equation 11-2 with history y(t)
= S(t) for t < t0.
ddesd function integrates with the classic
four-stage, fourth-order explicit Runge-Kutta method, and it controls
the size of the residual of a natural interpolant. It uses iteration
to take steps that are longer than the delays.
Delay differential equations of neutral type involve delays in y ′ as well as y:
The delays in the solution must satisfy dyi(t,y) ≤ t. The delays in the first derivative must satisfy dypj(t,y) < t so that y ′ does not appear on both sides of the equation.
and the output from any of the DDE solvers to evaluate the solution
at specific points in the interval of integration. For example,
= deval(sol, 0.5*(sol.x(1) + sol.x(end))) evaluates the
solution at the midpoint of the interval of integration.
When you solve a DDE, you approximate the solution on an interval [t0,tf] with t0 < tf. The DDEs show how y ′(t) depends on values of the solution (and possibly its derivative) at times prior to t. For example, Equation 11-1 shows that y ′(t0) depends on y(t0 – τ1),…,y(t0 – τk) for positive constants τj. Because of this, a solution on [t0, tk] depends on values it has at t ≤ t0. You must define these values with a history function, y(t) = S(t) for t <t0.
Generally, the first derivative of the solution has a jump at the initial point. This is because the first derivative of the history function, S(t), generally does not satisfy the DDE at this point. A discontinuity in any derivative of y(t) propagates into the future at spacings of τ1,…, τk when the delays are constant, as in Equation 11-1. If the delays are not constant, the propagation of discontinuities is more complicated. For neutral DDEs of the form given by Equation 11-1 or Equation 11-2, the discontinuity appears in the next higher order derivative each time it is propagated. In this sense, the solution gets smoother as the integration proceeds. Solutions of neutral DDEs of the form given by Equation 11-3 are qualitatively different. The discontinuity in the solution does not propagate to a derivative of higher order. In particular, the typical jump in y ′(t) at t0 propagates as jumps in y ′(t) throughout [t0, tf].
 Shampine, L.F. "Dissipative Approximations to Neutral DDEs." Applied Mathematics & Computation, Vol. 203, 2008, pp. 641–648.