Phase portraits of the standard map at 2 couplings

At its defaults it draws phase portraits of the standard map at 2 couplings. The standard map p → p + K sin θ, θ → θ + p, iterated 220 times from 12 starting points, at couplings of 0.6 and 1.3. Both coordinates run from 0 to 2π. An orbit that lies on a curve spanning the picture from left to right is an invariant circle, and nothing can cross it; an orbit that fills an area is chaotic; an orbit that circulates round a centre is trapped in a resonance island. At K = 0.6, orbits launched on p = 0 get no further than 1.55 in p, so a spanning curve is still there. At K = 1.3, orbits launched on p = 0 get no further than 10.78 in p, so a spanning curve is gone and transport is global. The point of the pair is that the change between them is not a change of character in any single orbit — chaotic orbits and regular ones coexist on both sides — but the loss of the barriers that kept the chaotic ones local.

kam-tori is one function in lib/figures/mechanics.js — motion, force, energy and rotation. 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 standard map p → p + K sin θ, θ → θ + p, iterated 220 times from 12 starting points, at couplings of 0.6 and 1.3. Both coordinates run from 0 to 2π. An orbit that lies on a curve spanning the picture from left to right is an invariant circle, and nothing can cross it; an orbit that fills an area is chaotic; an orbit that circulates round a centre is trapped in a resonance island. At K = 0.6, orbits launched on p = 0 get no further than 1.55 in p, so a spanning curve is still there. At K = 1.3, orbits launched on p = 0 get no further than 10.78 in p, so a spanning curve is gone and transport is global. The point of the pair is that the change between them is not a change of character in any single orbit — chaotic orbits and regular ones coexist on both sides — but the loss of the barriers that kept the chaotic ones local.

1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start.

The logistic map's third iterate close to one of the three places where it nearly meets the diagonal, at r = 3.828427 − 0.0003, on a window 0.024 wide. The solid curve stays above the diagonal by a gap of 6.1 × 10⁻⁴, measured by searching the curve; the dashed curve is the same iterate at the window's edge, where the gap closes and a stable three-cycle appears at the point of contact. The staircase is an orbit passing through the channel between the curve and the diagonal, taking 17 steps of the third iterate to cross this window — tiny steps in the middle, where the gap is narrowest, and larger ones at the ends. Halving the distance to the window halves the gap, 3.0 × 10⁻⁴, and the curvature at the channel is 34.1, so the longest passage is π divided by the square root of curvature times gap, 22 steps.

The mean length of the calm stretches, in iterates, against the distance below the setting at which the logistic map's three-cycle is born, on logarithmic axes, measured from runs of the map long enough to contain at least forty calms at each setting. At 1 × 10⁻⁸ the calms average 8125 iterates over 2444 of them; at 3 × 10⁻⁸ the calms average 4707 iterates over 4195 of them; at 1 × 10⁻⁷ the calms average 2547 iterates over 7684 of them; at 3 × 10⁻⁷ the calms average 1484 iterates over 12976 of them; at 1 × 10⁻⁶ the calms average 804 iterates over 23098 of them; at 3 × 10⁻⁶ the calms average 456 iterates over 33630 of them; at 1 × 10⁻⁵ the calms average 245 iterates over 31072 of them. A straight line fitted to the measurements has slope −0.506. The upper line is the longest passage through the channel, π divided by the square root of curvature times gap, with both measured on the map's third iterate, whose slope is −0.500. Both are −½ because the gap grows in proportion to the distance from the window and a creep through a parabolic channel takes a time inversely proportional to the square root of its gap: a hundred times closer to the window, ten times longer calms.

The lengths of 19620 calm stretches in a run of six million iterates at a distance of 1 × 10⁻⁵ below the window, as a histogram. The dashed line is the longest passage through the channel, 358 iterates, computed from the gap and the curvature of the map's third iterate. The distribution is not centred on a typical value: it rises towards that longest passage and piles up just below it, with its tallest bin between 329 and 343 iterates. A burst that happens to throw the orbit back near the entrance of the channel gives a long calm, and one that drops it part-way through gives a short one, so calms longer than the channel allows are absent: the longest seen, 342, is also the length ninety-nine calms in a hundred fall short of.

The Lyapunov exponent of the logistic map — the average rate at which neighbouring orbits separate — against its setting, across the edge of the three-cycle window at 3.828427, one plus the square root of eight, from 60,000 iterates at each of 161 settings. Below the edge it is positive and the orbit is chaotic; across the edge it drops below zero, and inside the window the stable three-cycle pulls neighbouring orbits together. Close to the edge the exponent is measured separately at 7 settings from 1 × 10⁻⁸ to 1 × 10⁻⁵ below it: 1 × 10⁻⁸ gives 0.0033, 3 × 10⁻⁸ gives 0.0058, 1 × 10⁻⁷ gives 0.0101, 3 × 10⁻⁷ gives 0.0174, 1 × 10⁻⁶ gives 0.0334, 3 × 10⁻⁶ gives 0.0537, 1 × 10⁻⁵ gives 0.0914, and a line fitted on logarithmic axes has slope 0.48. The chaos does not switch off; it fades as the square root of the distance, because separation happens only in the bursts and bursts become rarer as the calms lengthen. The law is a limit: between 3 × 10⁻⁵ and 10⁻⁴ below the edge the same exponent grows only as the 0.32 power, because there the calms are too short for the channel's parabola to dominate them.

physicscheck asserts something about kam-tori 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.