Illustrated Physics
physics, drawn
Concept

Attractor — where it appears

The set of states a dissipative system settles onto after its transients have died, which may be a point, a closed loop, or a fractal. Which attractor a system reaches depends on where it started, and the boundaries between the basins can be arbitrarily intricate.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The set an orbit that never repeats settles onto. 24,000 successive positions of one orbit of the map x' = 1 − 1.4x² + y, y' = 0.3x, after five hundred steps of transient have been discarded. Nearby points separate at e^0.4188 per step, so the orbit is unpredictable in the way the rung below measures; and every one of the 24,000 points lies inside a box 2.558 by 0.767, a diagonal of 2.670, so it is going nowhere. Those two statements are not compatible with a smooth stretching: something has to bring the separated points back, and the bringing back is the visible fold at the left-hand end. The curve is not a curve. Every strand of it is a bundle of strands at any magnification, which is what an area contraction of 0.3 per step leaves behind when the stretching along the other direction is e^0.419. The 24,000 points paint 7,352 distinct marks at the resolution this is drawn at, which is itself a measurement of how little of the plane the set occupies.

The fold that has to be there

Two trajectories that separate exponentially, in a region they can never leave, are being asked to do two incompatible things. The resolution is that the motion is folded back on itself over and over, and the object that survives infinitely many foldings is neither a curve nor a patch of surface — it has a dimension between the two, and the number can be measured two entirely different ways.

mechanics · Chaos Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time.

The map a dripping tap turns out to be

A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely.

mechanics · Chaos Long stretches of order, broken without warning. 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 calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.

mechanics · Chaos

Named alongside it

The objects these essays reach for when they reach for this one.

ChaosLyapunov exponentBifurcationBox countingDeterminismDissipationFractal dimensionIntermittencyJosephson effectKaplan yorkeLogistic mapNonlinearity

All concepts