The main control is a product.
For β ≠ 0, set u = βw. The rule becomes u′ = Aβ sin(x + φ + u). Aβ governs the normalized update. Equal products need matched normalized initial conditions to give identical trajectories.
FEEDBACK & PERSISTENCE OF STRUCTURE
Can a simple rule
remember?
We repeat a calculation, feeding each answer back into the next step. Try writing a bit and disturbing it. Then feed in a sequence and test how much of the recent past remains readable.
These experiments explore a larger question: what allows something recognizable to persist through change?
The wave lab below explores the rule behind these demos. Each point’s current height helps determine its next height. Compare two starting waves to see whether their differences survive.
THE WAVEFORM
Shape over position
ITERATION
Initial sineCurrentNearby start
One-step changeRMS · current vs. previous
Two-step changeRMS · current vs. two steps ago
Distance between runsRMS · current curves
Small one-step change suggests settling. Small two-step change with larger one-step change suggests alternation. A small distance between runs means their current curves nearly coincide. RMS combines the gaps across sampled positions into one overall measure.
FOLLOW ONE POSITION
A local history
—Last 500 states at most
Each sampled position evolves independently.
EXPERIMENT 02 · PERSISTENCE OVER TIME
A shape can repeat without standing still.
Compare measured outcomes below, at, and above the feedback threshold. Each count describes sampled positions in a finite run.
| Experiment / gain Aβ | Fixed-like | Two-cycle-like | Unresolved | Inspect |
|---|
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What these labels mean
A position is fixed-like when every one-step difference in the last 100 updates is at most 10⁻⁷. It is two-cycle-like when every two-step difference meets that tolerance and every one-step difference exceeds it. Other positions are unresolved. We require a full window of valid two-step differences, so classifications first become available at iteration 101. These labels describe the primary trajectory; they do not prove an attractor is stable or diagnose chaos.
CHANGE THE BEGINNING, KEEP THE RULE
Does the initial shape leave a trace?
Compare sine with an inverted or flat start, using A = 1, φ = 0 and 257 positions. Distance is the RMS difference between their states. The final column advances the comparison one extra step everywhere.
| Feedback β | Comparison | Initial distance | At 10,000 | Comparison +1 step | Inspect |
|---|
The flat start is a control: with these settings its first update is sin(x), so it follows the original wave exactly one step behind. A large same-time distance can therefore be entirely a timing difference. The inverted start tests a different beginning; small shifted distance is evidence of alignment in this run, not a proof about limiting behavior or usable memory.
FEEDBACK & PERSISTENCE OF STRUCTURE
When does persistence become memory?
A lasting shape can be recreated by the rule even as differences between starting waves disappear. To test usable memory, we need something to write and something to read back.
Try retaining a bit, then recovering recent inputs. These experiments give persistence a measurable job.
The sine feedback rule has established mathematical relatives. A research contribution would need a specific result: a useful persistence measure, a new bound, or a distinct effect that survives comparison with known models.
Persistence can come from attraction.
When |Aβ| is below 1, every starting state converges to the same profile at each position. A shape can persist even as differences between starting shapes disappear.
A connection to existing work.
Our fixed-position equation is a special case of the sine-based family in Sprott’s Strange Attractors, §7.4 (1993). This is a bounded real-valued map, with independently evolving positions.