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). 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:

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

The delay differential equations that we study are typically autonomous delay differential equations: the nature of the expression for $d(x(t))/dt$ is invariant under translation of $t$ by any constant.