Delay differential equation

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 $\tau \in \mathbb {R}$, 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 $t$.