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Solve delay differential equations (DDEs) with constant delays




Function handle that evaluates the right side of the differential equations . The function must have the form

  • dydt = ddefun(t,y,Z)

where t corresponds to the current , y is a column vector that approximates , and Z(:,j) approximates for delay = lags(j). The output is a column vector corresponding to .

Vector of constant, positive delays .
Specify history in one of three ways:
  • A function of such that y = history(t) returns the solution for as a column vector
  • A constant column vector, if is constant
  • The solution sol from a previous integration, if this call continues that integration
Interval of integration as a vector [t0,tf] with t0 < tf.
Optional integration argument. A structure you create using the ddeset function. See ddeset for details.


sol = dde23(ddefun,lags,history,tspan) integrates the system of DDEs

on the interval , where are constant, positive delays and . ddefun is a function handle. See Function Handles in the MATLAB Programming documentation for more information.

Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function odefun, if necessary.

dde23 returns the solution as a structure sol. Use the auxiliary function deval and the output sol to evaluate the solution at specific points tint in the interval tspan = [t0,tf].

The structure sol returned by dde23 has the following fields.

Mesh selected by dde23
Approximation to at the mesh points in sol.x.
Approximation to at the mesh points in sol.x
Solver name, 'dde23'

sol = dde23(ddefun,lags,history,tspan,options) solves as above with default integration properties replaced by values in options, an argument created with ddeset. See ddeset and Initial Value Problems for DDEs in the MATLAB documentation for details.

Commonly used options are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector of absolute error tolerances 'AbsTol' (all components are 1e-6 by default).

Use the 'Jumps' option to solve problems with discontinuities in the history or solution. Set this option to a vector that contains the locations of discontinuities in the solution prior to t0 (the history) or in coefficients of the equations at known values of after t0.

Use the 'Events' option to specify a function that dde23 calls to find where functions

vanish. This function must be of the form

and contain an event function for each event to be tested. For the kth event function in events:

If you specify the 'Events' option and events are detected, the output structure sol also includes fields:

Row vector of locations of all events, i.e., times when an event function vanished
Matrix whose columns are the solution values corresponding to times in sol.xe
Vector containing indices that specify which event occurred at the corresponding time in sol.xe


This example solves a DDE on the interval [0, 5] with lags 1 and 0.2. The function ddex1de computes the delay differential equations, and ddex1hist computes the history for t <= 0.

This code evaluates the solution at 100 equally spaced points in the interval [0,5], then plots the result.

ddex1 shows how you can code this problem using subfunctions. For more examples see ddex2.


dde23 tracks discontinuities and integrates with the explicit Runge-Kutta (2,3) pair and interpolant of ode23. It uses iteration to take steps longer than the lags.

See Also

ddeget, ddeset, deval, function_handle (@)


[1]  Shampine, L.F. and S. Thompson, "Solving DDEs in MATLAB," Applied Numerical Mathematics, Vol. 37, 2001, pp. 441-458.

[2]  Kierzenka, J., L.F. Shampine, and S. Thompson,"Solving Delay Differential Equations with DDE23," available at

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