Delay differential equation

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In mathematics, delay differential equations (DDEs) are a type of differential equation, in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.

A general form of the time-delay differential equation for $x(t)\in R^n$ is

$\frac{d}{dt}x(t)=f(t,x(t),x_t),$

where $x_t=\{x(\tau):\tau\leq t\}$ represents the trajectory of the solution in the past. In this equation, $f$ is a functional operator from $R\times R^n\times C^1$ to $R n$

[edit] Examples

Continuous delay

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)d\mu(\tau)\right)$

Discrete delay

$\frac{d}{dt}x(t)=f(t,x(t),x(t-\tau_1),\ldots,x(t-\tau_n))$ for $\tau_1>\ldots>\tau_n\geq 0$ .

Another example is given by

$\frac{d}{dt}x(t) = ax(t) + bx(\lambda t),$

where a, b and λ are constants and 0 < λ < 1. This equation, often in some more general form, is called the pantograph equation (after the pantographs on trains).

[edit] Reduction to ODE

In some cases, delay differential equations are equivalent to a system of ordinary differential equations.

Example 1 Consider an equation

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}d\tau\right)$

Introduce $y(t)=\int_{-\infty}^0x(t+\tau)e^{\lambda\tau}d\tau$ to get a system of ODEs

$\frac{d}{dt}x(t)=f(t,x,y),\quad \frac{d}{dt}y(t)=x-\lambda y$

Example 2 An equation

$\frac{d}{dt}x(t)=f\left(t,x(t),\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)d\tau\right)$

is equivalent to

$\frac{d}{dt}x(t)=f(t,x,y),\quad \frac{d}{dt}y(t)=\cos(\beta)x+\alpha z,\quad \frac{d}{dt}z(t)=\sin(\beta) x-\alpha y$

where

$y=\int_{-\infty}^0x(t+\tau)\cos(\alpha\tau+\beta)d\tau,\quad z=\int_{-\infty}^0x(t+\tau)\sin(\alpha\tau+\beta)d\tau$

[edit] The Characteristic Equation

Solutions of DDEs can be found by studying the 'characteristic equation', as for ODEs. However, for DDEs, the characteristic equation can have an infinity of solutions, making analysis hard. Consider, for example, the following equation:

$\frac{d}{dt}x(t)=-x(t-1).$