Laboratory
A Loop That Learns
Adjust gain and delay in a simple feedback loop to see when correction steadies a system and when overreaction makes it run away.
A dimensionless recurrence illustrating feedback, delay, and stability.
In cybernetics and in daily life, feedback that arrives with a gentle gain helps a process find its mark, while delayed feedback coupled with an aggressive gain can send a system into wild oscillation. This instrument runs the dimensionless recurrence x[t+1] = x[t] + gain × (target − x[max(0, t − delay)]), starting from x[t] = 0 for t ≤ 0. If you choose values that cause the sequence to diverge, the chart shows the runaway plainly rather than hiding it.
x[t+1] = x[t] + gain * (target - x[max(0, t - delay)])
Dimensionless recurrence with initial history x[t] = 0 for t ≤ 0. Identical inputs always produce the same trajectory.
| t | x[t] |
|---|---|
| 0 | 0.0000 |
| 1 | 0.3500 |
| 2 | 0.7000 |
| 3 | 1.0500 |
| 4 | 1.2775 |
| 5 | 1.3825 |
| 6 | 1.3650 |
| 7 | 1.2679 |
| 8 | 1.1340 |
| 9 | 1.0063 |
| 10 | 0.9125 |
| 11 | 0.8656 |
| 12 | 0.8634 |
| 13 | 0.8940 |
| 14 | 0.9411 |
| 15 | 0.9889 |
| 16 | 1.0260 |
| 17 | 1.0466 |
| 18 | 1.0505 |
| 19 | 1.0414 |
| 20 | 1.0251 |
| 21 | 1.0074 |
| 22 | 0.9929 |
| 23 | 0.9841 |
| 24 | 0.9816 |