Delay Differential Equations

where x t ( ) x (t ) and 0. Observe that xt ( ) with 0 represents a portion of the solution trajectory in a recent past. Here, f is a functional operator that takes a time input and a continuous function xt ( ) with 0 and generates a real number (dx (t )/dt ) as its output. A well-known example of a delay differential equation is the Hutchinson equation, or the discrete delay logistic equation, x rx (1 x (t )/K ). Some DDEs can be conveniently solved in a stepwise fashion. In fact, the Hutchinson equation can be rewritten as (ln x ) r (1 x (t )/K ), which can be used to solve for x for 0 t . Some DDEs, such as x (t ) rx (t )[1 a ∫ 0 e as x (t s )ds/K ], a 0, are in fact a system of ordinary diffential equations (ODEs) in disguise. This can be seen by letting y (t ) a ∫ 0 e as x (t s )ds and noticing that y a(x (t ) y (t )), which yields a system of ODEs x (t ) rx (t )(1 y (t )/K ); y a(x (t ) y (t )). Indeed, an integro-differential equation of the form x (t ) f (t, x (t )) ∫ 0 k (s )g (x (t s ))ds with initial condition xt( ) where 0 is equivalent to a system of ODEs with initial condition if k is a linear combination of functions eat, teat, t 2eat, . . . , tmeat, where a is a real number and m is a positive integer. The method of reducing such a delay differential equation into a system of ODEs is called the linear chain trick. DELAY DIFFERENTIAL EQUATIONS