Delay differential equation

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Definition

The notion of delay differential equation (abbreviated DDE) is a variant of the notion of differential equation (in other words, delay differential equations are not (ordinary) differential equations).

First-order first-degree case

If we denote the dependent variable by x and the independent variable by t (Which we think of as time), the first-order first-degree case is:

\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)

General case

The general case of a delay differential equation is of the form:

F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t) = 0

Note on autonomous case

The delay differential equations that we study are typically autonomous delay differential equations: an equation in the general form above is autonomous if, for any \tau \in \R, the function F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t) is invariant under replacing x(t) by the function t \mapsto x(t - \tau). Intuitively, what this means is that t does not appear explicitly in F, and all the behavior at previous points is specified in terms of how much earlier they were than t.