RecallWeave · Dynamics lab
When correction overshoots
A correction can settle a system, make it alternate, or send it farther away. Change one rule, compare two starting states, and explain what the next update will do.
Start with a behavior
Set the experiment
Keep the rule constant during each run. The second run starts at x[0] + δ, with every other input unchanged.
The same rule, two starting states
The vertical scale follows both recorded trajectories and the reference. The update index n has no specified duration in seconds.
- Current state x[n]
- Reference error r − x[n]
- Applied correction u[n]
- Next state x[n+1]
Try · predict · explain
Explore before answering
- Predict the first update. With the default rule and x[0] = 0, calculate the correction, then the next state. Move the inspection control to check.
- Increase correction k. Compare gentle correction, alternating recovery, the repeating boundary, and growth. Explain each change using q.
- Challenge a flat line. Open “A flat run can hide instability.” One run stays at zero. What does the different starting state reveal?
- Separate two outcomes. Reset, then change d to −0.25. Does q change? Does the fixed point equal the reference?
Open RecallWeave, choose a local JSON lesson, and select this download. Its six connected concepts include explanations and transfer prompts. The download contains the course questions and their authored explanations; it does not include your experiment.
Read the rule
Why q predicts the separation
Substitute the correction to get x[n+1] = q x[n] + c, where c = k r + d. Subtract the two runs: their common constant input cancels. In exact arithmetic, a starting difference δ becomes δ qⁿ after n updates.
A fixed point can miss the reference
If q ≠ 1, the fixed point is x* = c / (1 − q). At the default settings it is 2/3, while the reference is 1. A stable recurrence can therefore retain a nonzero reference error. A fixed point exists even when perturbations grow.
The boundary needs its own name
If |q| < 1, deviations decay. If |q| > 1, nonzero deviations grow. At q = −1 they repeat with alternating sign. At q = 1 the separation persists; c = 0 makes every state fixed, while c ≠ 0 causes drift. These boundary cases do not have decaying perturbations.
What this experiment establishes
This is a dimensionless scalar recurrence with constant coefficients, no delay, no saturation and no noise. A finite trace illustrates its behavior; the geometric multiplier supplies the all-update argument. Actual systems and learner outcomes require their own evidence.
Why the last digits can differ
Controls use exact quarter increments, and the computer iterates with ordinary finite-precision numbers. Very small differences can round away when two state values become equal. The table and CSV include both computed separation and the analytical value δ qⁿ.
Inspect all recorded states
There are N + 1 state rows for N updates. The final state has no applied correction or recorded next state. Export includes both runs, settings and full numeric values.
| n | x[n] | Different start | r − x[n] | u[n] | x[n+1] | Computed separation | δ qⁿ |
|---|