Where modulating a system sets it going

At its defaults it draws where modulating a system sets it going. The regions of the modulation plane in which an oscillator with a damping ratio of 0.02 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 8.0%, against 40.0% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything.

parametric-tongues is one function in lib/figures/waves.js — travelling, standing, adding and shifting. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The regions of the modulation plane in which an oscillator with a damping ratio of 0.02 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 8.0%, against 40.0% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything.

The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

The least shaking amplitude that holds a pendulum upside down, against how fast the pivot is shaken. The dots are measured: at each rate the exact equation of motion is integrated from a start a quarter of a radian off the inverted position, and the amplitude is bisected until the pendulum stops falling over. The curve is the averaged theory, aΩ = √(2gL), which involves no integration at all. The two agree to 3 per cent over rates of 8, 14, 20, 30 times the natural frequency, and the disagreement grows toward the slow end, which is where the separation of timescales that the averaging assumes is weakest. The product is what matters rather than either factor: shaking twice as fast needs half the amplitude, and it is the shake's *speed* that has to beat the speed of a free fall through the pendulum's length. There is an upper edge as well, marked by the second row of dots at 0.39, 0.36, 0.35, 0.35 of the length, and it is a different mechanism: shaken that hard the fast motion is no longer small, the averaging that produced the criterion stops applying, and the pendulum is thrown out by the parametric instability this generator's other modes are about. Stability is a band and not a threshold.

The regions of the modulation plane in which an oscillator with a damping ratio of 0.005 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 2.0%, against 20.1% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything.

Two runs of the same oscillator, damping ratio 0.02, both started from the same small displacement and both modulated at 2× their natural frequency. The growing one has its stiffness modulated by 22%; the flat one by 6%, and it dies away exactly as if nothing were being done to it. There is no force in either equation — the right-hand side is zero, so standing still is always a solution — and what the modulation changes is whether standing still is stable. The envelope of the growing case is an exponential of rate 0.0306 per unit time, reaching 204× its starting amplitude in 26 natural periods. A driven oscillator, by contrast, responds to any force however small, and settles rather than growing.

A particle in a potential that is a saddle — pushing it out along one axis and in along the other — whose sign is reversed at a steady rate. Four strengths are integrated, at q of 0.2, 0.45, 0.7, 0.95. The bounded curves show the two motions such a trap always has: a fast, small wobble at the switching rate, called the micromotion, and a slow oscillation of the average position, which is the trapping proper. The slow motion is what an effective potential describes, and the trap is real: a static saddle cannot hold anything, and this one holds it without ever having a minimum. Above q = 0.908 — measured here by bisecting on the integration, against Mathieu's 0.908 — the particle leaves and does not come back. That is the same criterion as the shaken pendulum's, in the same equation, with the roles of the constant and the alternating term exchanged.

physicscheck asserts something about parametric-tongues that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.