Delayed feedback

**Feedback** refers to a situation where the evolution of a system is influenced by its own state. For reasons of mathematical tractability, most textbook models of feedback assume instantaneous self-action. For instance, here’s the simplest linear model of negative feedback:

${d\over d t}x(t) = - \alpha x(t).$

It is well known that the solutions of this equation decay exponentially to zero with a characteristic time $\tau = 1/\alpha$. More generally, one can describe a generic feedback situation by a model of the form

${d\over d t}x(t) = f\bigl(x(t)\bigr).$

This is very general as one can consider $x$ to be vector valued and to include state variables and their rates of change if necessary for a full description.

**Delayed feedback** refers to the situation where the self-action is not instantaneous but takes some time. For instance, modifying the above with a given constant delay $\Delta$:

${d\over d t}x(t) = f\bigl(x(t-\Delta)\bigr).$

This equation has the same constant solutions as before, characterised by $f\bigl(x(t)\bigr) = 0$ $\forall t$, but time-varying solutions now overshoot? the constant ones. This is because, when $f(x)=0$, the system will continue moving in the direction dictated by $f(x-\Delta)\neq 0$. In fact, in the simple 1-dimensional model, crossing the stable level indicates that the system will reverse its direction of motion after a delay $\Delta$. Depending on the speed with which the stable level is crossed, successive oscillations may grow or decay, so the system may be unstable if it moves very fast.

As a consequence of delay and overshoot, the delayed-feedback system is qualitatively completely different form the instantaneous-feedback one:

- The space of solutions of the original equation was one-dimensional - all solutions were uniquely determined by their initial conditions at a given time, $x(t_0)$. By contrast, the space of solutions of the delayed equation is infinite-dimensional, as a solution is only uniquely determined by knowing $x(t)$ for all $t\in[t_0-\Delta, t_0]$.
- The delayed equation is generally unstable, even if the case with no delay is stable.

We can see this in detail in the above one-dimensional model of delayed feedback,

${d\over d t}y(t) = f\bigl(y(t-\Delta)\bigr).$

Let $y_0$ be such that $f(y_0) = 0$. Then, $y(t) = y_0$ $\forall t$ is a possible solution. To analyse its stability we approximate $f(y)\approx f(y_0) + f'(y_0)(y-y_0)$ and, since $f(y_0) = 0$,

${d\over d t}\bigl[y(t)-y_0\bigr] = f'(y_0)\bigl(y(t-\Delta)-y_0\bigr).$

Letting $x = y - y_0$, we obtain the simple model of delayed linear feedback with delay, for $\alpha = -f'(x_0)$.

${d\over d t}x(t) = - \alpha x(t-\Delta).$

Now normalize the system, measuring time in units of $\tau$. Defining $t\colon = \tau s$ and $y(s) \colon = x(\tau s)$,

${d\over d s}y(s) = - y(s - \Delta/\tau).$

In other words, we can just consider the case $\alpha = 1$ if we express $\Delta$ in units of $\tau$. We simply have

${d\over d t}x(t) = - x(t-\Delta).$

We now discretize the system with a time step $\Delta/n$, for arbitrary but relatively large $n$ (after all, we assume $\Delta$ is “macroscopic” while derivatives involve “infinitesimal” times). We approximate

$x(t) \approx x(t-\Delta/n) + {\Delta\over n} x'(t).$

Then,

$x(t) - x(t-\Delta/n) + {\Delta\over n} x(t-\Delta) \approx 0,$

which, sampling $x(t)$ when $t$ is a multiple of $\Delta/n$, yields a finite-difference equation with characteristic polynomial

$z^n - z^{n-1} + {\Delta\over n} \approx 0$

corresponding to the ansatz $x(\Delta m/n) \approx z^m$.

It can be seen that not only the delayed equation has an infinite collection of approximate solutions, but also for sufficiently large $\Delta$ and $n$ one can find approximate solutions with $\left\vert z\right\vert \gt 1$ making the autonomous system unstable.

In the numerical experiments reported below, $\Delta\approx 1.65$ results in a stable amplitude of oscillation whereas larger values lead to instability and smaller values to stability, but with slower convergence to zero than in the undelayed case.

The following picture represents the values of $z$ for $\Delta = 1,2,4$ and $n = 1,\ldots, 20$:

As $n$ grows for any $\Delta \gt 0$, the solutions of the characteristic equation approach the unit circle and cover the whole circle ever more densely, which means that the driven equation

${d\over d t}x(t) + x(t-\Delta) = f(t).$

will resonate with any periodic component of the driver $f(t)$, in particular with any fundamental or numerical noise components, leading to secular growth of the response.

The following picture represents some solutions of the autonomous (unforced) delayed differential equation for various values of $\Delta$:

In each case, the initial condition is equal to $1$ for a length of time equal to $\Delta$ and the equation is integrated in steps of length $\Delta/n$ for a total time of $10\Delta$.

Delayed Feedback Example R Code #2

- A. B. Pippard,
*Response and Stability: an Introduction to the Physical Theory*, Cambridge University Press, Cambridge, 1985.

For stochastic aspects, see:

- Stochastic delay differential equation, Azimuth Library.

category: mathematical methods