Delay Differential Equations (DDEs) provide a context to study systems dependent on past states. To illustrate, consider the general ODE:

In order to specify a single unique solution to this equation under sufficiently nice properties, we need a point on the curve .

DDEs require additional information.

Consider the Mackey-Glass equation:

where and

This equation describes the density of mature, circulating white blood cells. It requires initial data on the interval the to provide a solution on .

The State Space:

Let’s focus our attention on a simpler DDE:

with constant initial condition on the relevant interval.

Our aim is to describe a space that contains all necessary information to determine a solution of a DDE.

Clearly, we need an initial condition to start thinking about a solution. Take on .

Define as all pairs . In order to determine future values of the DDE we need for all . This is is equivalent to knowing for

Take the element of to be the State Space for the DDE. The trajectory of the solution will be the curve in .

Now take for example a similar curve to the above:

on

The following matlab code can be used to create the above plot of the State Space:

lags = 1 ; hold on; while lags > 0 sol = dde23(@ddex1de,lags,@ddex1hist,[ 0 , 10 ]); plot3(sol.x + lags, sol.y, lags * ones ( length (sol.x))); lags = lags - 0.01 end function dydt = ddex1de (t,y,Z) ylag1 = Z(:, 1 ); dydt = - 0.75 * ylag1( 1 ); end function S = ddex1hist (t) S = ones ( 1 , 1 ); end

Framing DDEs in terms of a state space paves the way for consideration of the uniqueness and existence of solutions, which will follow in a later post. Thanks for reading.