# Delay differential equation

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function :
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
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. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:
* Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, and communication networks that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering.
* Delay systems are still resistant to many classical controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations.
* Voluntary introduction of delays can benefit the control system.
* In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).
A general form of the time-delay differential equation for $x(t)\in \mathbb {R} ^{n}$ 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 $\mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})$ to $\mathbb {R} ^{n}.$

## Examples

* Continuous delay ${\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )\,d\mu (\tau )\right)$
* Discrete delay ${\frac {d}{dt}}x(t)=f(t,x(t),x(t-\tau _{1}),\dots ,x(t-\tau _{m}))$ for $\tau _{1}>\dots >\tau _{m}\geq 0.$
* Linear with discrete delays ${\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\dots +A_{m}x(t-\tau _{m})$ where $A_{0},\dotsc ,A_{m}\in \mathbb {R} ^{n\times n}$.
* Pantograph equation ${\frac {d}{dt}}x(t)=ax(t)+bx(\lambda t),$ where a, b and λ are constants and 0 < λ < 1. This equation and some more general forms are named after the pantographs on trains.

## Solving DDEs

DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay ${\frac {d}{dt}}x(t)=f(x(t),x(t-\tau ))$
with given initial condition $\phi \colon [-\tau ,0]\to \mathbb {R} ^{n}$. Then the solution on the interval $[0,\tau ]$ is given by $\psi (t)$ which is the solution to the inhomogeneous initial value problem ${\frac {d}{dt}}\psi (t)=f(\psi (t),\phi (t-\tau )),$ with $\psi (0)=\phi (0)$. This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.

### Example

Suppose $f(x(t),x(t-\tau ))=ax(t-\tau )$ and $\phi (t)=1$. Then the initial value problem can be solved with integration,
$x(t)=x(0)+\int _{s=0}^{t}{\frac {d}{dt}}x(s)\,ds=1+a\int _{s=0}^{t}\phi (s-\tau )\,ds,$
i.e., $x(t)=at+1$, where the initial condition is given by $x(0)=\phi (0)=1$. Similarly, for the interval $t\in [\tau ,2\tau ]$ we integrate and fit the initial condition,
${\begin{aligned}x(t)=x(\tau )+\int _{s=\tau }^{t}{\frac {d}{dt}}x(s)\,ds&=(a\tau +1)+a\int _{s=\tau }^{t}\left(a(s-\tau )+1\right)ds\\&=(a\tau +1)+a\int _{s=0}^{t-\tau }\left(as+1\right)ds,\end{aligned}}$
i.e., x(t)=(aτ+1)+a(t−τ)( 1/2 a(t−τ)+1).{\textstyle x(t)=(a\tau +1)+a(t-\tau )\left({\frac {1}{2}}{a(t-\tau )}+1\right).}

## Reduction to ODE

In some cases, differential equations can be represented in a format that looks like delay differential equations.
* Example 1 Consider an equation ${\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }\,d\tau \right).$ Introduce $y(t)=\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }\,d\tau$ 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 ${\frac {d}{dt}}x(t)=f\left(t,x(t),\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )\,d\tau \right)$ 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 $y=\int _{-\infty }^{0}x(t+\tau )\cos(\alpha \tau +\beta )\,d\tau ,\quad z=\int _{-\infty }^{0}x(t+\tau )\sin(\alpha \tau +\beta )\,d\tau .$

## The characteristic equation

Similar to ODEs, many properties of linear DDEs can be characterized and analyzed using the characteristic equation. The characteristic equation associated with the linear DDE with discrete delays ${\frac {d}{dt}}x(t)=A_{0}x(t)+A_{1}x(t-\tau _{1})+\dots +A_{m}x(t-\tau _{m})$is the exponential polynomial given by$\det(-\lambda I+A_{0}+A_{1}e^{-\tau _{1}\lambda }+\dotsb +A_{m}e^{-\tau _{m}\lambda })=0.$
The roots λ of the characteristic equation are called characteristic roots or eigenvalues and the solution set is often referred to as the spectrum. Because of the exponential in the characteristic equation, the DDE has, unlike the ODE case, an infinite number of eigenvalues, making a spectral analysis more involved. The spectrum does however have some properties which can be exploited in the analysis. For instance, even though there are an infinite number of eigenvalues, there are only a finite number of eigenvalues in any vertical strip of the complex plane.
This characteristic equation is a nonlinear eigenproblem and there are many methods to compute the spectrum numerically. In some special situations it is possible to solve the characteristic equation explicitly. Consider, for example, the following DDE: ${\frac {d}{dt}}x(t)=-x(t-1).$ The characteristic equation is $-\lambda -e^{-\lambda }=0.$ There are an infinite number of solutions to this equation for complex λ. They are given by $\lambda =W_{k}(-1),$ where W$_k$ is the kth branch of the Lambert W function, so: $x(t)=x(0)\,e^{W_{k}(-1)\cdot t}.$

## Another example

The following DDE: ${\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).$
Have as solution in $\mathbb {R}$ the function: $u(t)={\begin{cases}F(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}$ with $F(t)$ the Fabius function.

## Applications

* Dynamics of diabetes
* Epidemiology
* Population dynamics
* Classical electrodynamics