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 $R^n.\,$

1 Examples
2 Reduction to ODE
3 The characteristic equation
4 External links

[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 linear DDEs can be studied by analyzing the characteristic equation, similarly to the ODEs. However, for DDEs, the characteristic equation can have continuum of solutions, making the spectral analysis hard. Consider, for example, the following equation:

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