odeDelay

ODE delay definition

Since R2025a

Description

An odeDelay object defines delays introduced in a differential equation and the history while solving a delay differential equation (DDE) using an ode object.

Create an ode object to represent an ODE problem, and specify an odeDelay object as the value of the DelayDefinition property to define the delays for the DDE.

Creation

Syntax

D = odeDelay

D = odeDelay(PropertyName=Value)

Description

D = odeDelay creates an odeDelay object with empty properties.

D = odeDelay(PropertyName=Value) specifies one or more property values using name-value arguments. For example, D = odeDelay(History=[0.1; 0.5]) defines the solution history [0.1; 0.5] for a DDE.

Properties

History — Solution history

function handle | vector

Solution history at t ≤ t₀, specified as a function handle or vector.

If you specify a handle to a function y = solhist(t) or y = solhist(t,p), then the function must return a column vector for a scalar t and an array p that represents the parameters specified by the Parameters property of the ode object.

To define a constant solution history, specify a vector.

Example: @ddeHist

Example: [1; 0.4; 0.2]

Data Types: single | double | function_handle

ValueDelay — Solution delays

function handle | vector

Solution delays, specified as a function handle that returns a vector of delays in y, or as a vector of constant delays.

If you specify ValueDelay as a function handle, the syntax must be dy = vdelay(t,y) or dy = vdelay(t,y,p), where the arguments are defined as described in this table.

Argument	Description
t	A scalar value that represents the current value of time t
y	A vector that represents y(t). The size of this vector is n-by-1, where n is the number of equations in the system you want to solve.
p	An array that represents the parameters specified by the Parameters property of the ode object
dy	A vector, returned by the vdelay function, whose values are the solution delays. The size of this vector must be p-by-1, where p is the number of solution delays in the equation. Each element of dy must be less than or equal to t.

To define constant solution delays of the form dy_i = t – τ_i, specify ValueDelay as a vector, where ValueDelay(i) = τ_i. Each value in this vector must be greater than or equal to zero.

Data Types: single | double | function_handle

SlopeDelay — Derivative delays

function handle | vector

Derivative delays, specified as a function handle that returns a vector of delays in yp, or as a vector of constant delays.

If you specify SlopeDelay as a function handle, the syntax must be dyp = sdelay(t,y) or dyp = sdelay(t,y,p), where the arguments are defined as described in this table.

Argument	Description
t	A scalar value that represents the current value of time t
y	A vector that represents y(t). The size of this vector is n-by-1, where n is the number of equations in the system you want to solve.
p	An array that represents the parameters specified by the Parameters property of the ode object
dyp	A vector, returned by the sdelay function, whose values are the derivative delays. The size of this vector must be q-by-1, where q is the number of derivative delays in the equation. Each element of dyp must be less than t, with one exception—if you are solving an initial value DDE, the value of dyp can equal t at t = t0. For more information, see Initial Value Neutral Delay Differential Equations.

To define constant derivative delays of the form dyp_j = t – τ_j, specify SlopeDelay as a vector, where SlopeDelay(j) = τ_j. Each value in this vector must be greater than zero, with one exception—if you are solving an initial value DDE of neutral type, a value in SlopeDelay can equal zero at t = t₀. See Initial Value Neutral Delay Differential Equations for more information.

Data Types: single | double | function_handle

Examples

Solve Delay Differential Equation with Constant Delays

Solve this system of delay differential equations (DDEs) by creating an ode object to represent the problem.

$\begin{array}{l}{{\mathit{y}}_1}^{\prime }\left(\mathit{t}\right)=-2{\mathit{y}}_1\left(\mathit{t}-2\right)+{\mathit{y}}_2\left(\mathit{t}\right)\\ {{\mathit{y}}_2}^{\prime }\left(\mathit{t}\right)={\mathit{y}}_1\left(\mathit{t}\right)-2{\mathit{y}}_2\left(\mathit{t}-1\right)\end{array}$

Define the system of DDEs as a local function named ddefun.

function dydt = ddefun(t,y,Z)
ydelay1 = Z(:,1);
ydelay2 = Z(:,2);
dydt = [-2*ydelay1(1) + y(2);
        y(1) - 2*ydelay2(2)];
end

Specify the system of equations in the ODEFcn property of an ode object. Specify the time delays as a vector [2; 1] and the solution history as a vector [0.1; 0.5] in the DelayDefinition property by using an odeDelay object.

F = ode;
F.ODEFcn = @ddefun;
F.DelayDefinition = odeDelay(ValueDelay=[2; 1],History=[0.1; 0.5]);

Solve the system of equations over the time interval [0 10] by using the solve method. Plot the results.

S = solve(F,0,10);
plot(S.Time,S.Solution,"-o")

Version History

Introduced in R2025a