Delay Differential Equations I

Consider a differential equation of the form

$\begin{equation} \label{eq:dde} \frac{\mathrm{d}x}{\mathrm{d}t} = f(t, x(t), x(t - \tau)) \end{equation}$

for some locally Lipschitz function $f \colon \mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d$ and $\tau > 0$ . An equation of the form $\ref{eq:dde}$ is an example of a delay differential equation (DDE). The $x(t-\tau)$ term signifies that the value $x(t)$ of a solution depends on the earlier state at time $t - \tau$ . In applications we are generally interested in solutions defined for $t>0$ , and where we specify an initial function $\varphi \colon [-\tau, 0] \to \mathbb{R}^d$ representing the history of the system before our initial time $t = 0$ .

In this note we provide a naive method for solving DDEs of the type in equation $\ref{eq:dde}$ . While this method is of limited practical use, it provides an initial understanding into the behaviour of solutions to DDEs.

The method of steps

Suppose we are given equation $\ref{eq:dde}$ with history function $\varphi$ on $[-\tau, 0]$ . Once $\varphi$ is fixed, equation $\ref{eq:dde}$ reduces to the initial value problem

$\begin{equation*} \begin{split} \frac{\mathrm{d}x}{\mathrm{d}t} &= f(t, x(t), \varphi(t - \tau)), \quad t \in (0, \tau]\\ x(0) &= \varphi(0) \end{split} \end{equation*}$

Such an IVP can be solved explicitly on $[0, \tau]$ using the usual techniques from ODE theory. Once we know the value of $x$ on $[0, \tau]$ we obtain a new ODE

$\begin{equation*} \begin{split} \frac{\mathrm{d}\widetilde{x}}{\mathrm{d}t} &= f(t, \widetilde{x}(t), x(t - \tau)), \quad t \in (\tau, 2\tau] \\ \widetilde{x}(\tau) &= x(\tau) \end{split} \end{equation*}$

A function $\widetilde{x}$ solving this new system gives an extension of our previous solution to the interval $[\tau, 2\tau]$ . Repeating this process we can extend our initial solution as far in time as required. The recursive algorithm for solving DDEs described above is called the method of steps.

Example

Fix some constant $\theta_0 \in \mathbb{R}$ and consider the DDE

$\begin{equation*} \begin{split} \frac{\mathrm{d} x}{\mathrm{d}t} &= - cx(t - \tau), \quad t > 0 \\ x(t) &\equiv \theta_0 \quad \forall t \in [-\tau, 0] \end{split} \end{equation*}$

In light of the initial history function being constant this reduces to the ODE

$\begin{equation*} \frac{\mathrm{d}x}{\mathrm{d}t} -c\theta_0, \quad x(0) = \theta_0 \end{equation*}$

on the interval $[0, \tau]$ . The solution is given by

$\begin{equation*} x(t) = -c\theta_0 t + \theta_0 \end{equation*}$

Stepping forward to $[\tau, 2\tau]$ we obtain the new ODE

$\begin{equation*} \frac{\mathrm{d}x}{\mathrm{d}t} = -c\theta_0(t-\tau) + \theta_0, \quad x(\tau) = -c\theta_0\tau + \theta_0 \end{equation*}$

which in turn has solution

$\begin{equation*} x(t) = -c\theta_o(t^2 - \tau t) + \theta_0t - \theta_0\tau(c + 1) + \theta_0 \end{equation*}$

Repeating this process we can extend this solution as long as we like.

In Delay Differential Equations II we consider more powerful methods for solving a class of DDE that includes equations of the type $\ref{eq:dde}$ .

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