Phase Fold
y = sin(x + sin(2x + t)) · cos(t · 0.3 + x · 0.5)
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What it draws
Three sines are stacked here: one inside another, and that pair times a third. The outer sin(x + …) is a carrier of wavenumber 1, with crests 2π ≈ 6.28 apart, and it holds no t of its own, so alone it would stand still.
The inner sin(2x + t) is added to its phase. It swings the carrier back and forth by up to one radian, and lines of constant phase satisfy 2x + t = constant, so that swing drifts left at 1/2 = 0.5 units per second at Speed 1. The final factor cos(0.3t + 0.5x) is a slow envelope. Its zeros are π/0.5 ≈ 6.28 apart, two of them across ±6, and they slide left at 0.3/0.5 = 0.6 units per second, pinching the curve flat as they pass.
Why it folds
Differentiate the phase of the outer sine and the local wavenumber falls out: 1 + 2·cos(2x + t), a number that runs between −1 and 3. Where it is large the crests bunch up. Where cos(2x + t) < −0.5, which is a third of the axis at any instant, it is negative and the phase runs backwards. That is the fold: a crest and a trough are created together, drift apart, and are swallowed again a moment later.
This is frequency modulation, the trick of an FM radio carrier, and the Jacobi–Anger expansion names the parts. The modulated sine is a sum of plain waves of wavenumber 1 + 2n weighted by Bessel values, J₀(1) ≈ 0.765 on the carrier and J₁(1) ≈ 0.440 on its first pair of sidebands.
Not chaos
Nothing here is sensitive to its starting conditions, because there are none: the height at a point is a formula in x and t. The two clock rates, 1 and 0.3, stand in the ratio 10:3, so the picture repeats exactly every 20π ≈ 62.8 seconds. It looks unruly only because three rhythms of similar size lie over one another.
Try
- Take the modulation out: sin(x)·cos(t·0.3 + x·0.5) is a still carrier under the same drifting envelope.
- Put it back at half strength, sin(x + 0.4·sin(2x + t))·cos(t·0.3 + x·0.5): the local wavenumber is now 1 + 0.8·cos(2x + t), never negative, so the wave breathes but never folds.
- Then overdo it, sin(x + 3·sin(2x + t))·cos(t·0.3 + x·0.5): the wavenumber ranges from −5 to 7 and the folds crowd together.
- Top in the deck: seen from above, each fold shows as a fork running back through the ribbon.