The window that opens with a stutter

The widest white gap in the chaotic half of the logistic map’s bifurcation diagram is the period-three window, and its left edge is not like the edges of the grey bands. The bands merge gradually, widening until they touch. The window arrives all at once: on one side of a parameter the orbit wanders the whole interval, and on the other side it sits on three points and stays there.

The edge is at exactly $r = 1 + \sqrt 8 \approx 3.828427$. But “all at once” describes where the orbit ends up, not how it behaves on the way. Just below the edge, the orbit does something more interesting than wander. It stutters.

The shaded stretches are the orbit imitating a period-three cycle that does not exist at this parameter. They end without warning, the orbit sprays across the interval for a while, and then it drops back into another quiet stretch. The pattern is called intermittency, and its timing obeys a law exact enough to be measured.

A cycle made from a collision #

A cycle of three for $f$ is a fixed point of the third iterate $f^3$, which is a polynomial of degree eight with four humps. Its graph crosses the diagonal $y = x$ at the two fixed points of $f$ itself, and whether it crosses anywhere else is the question of whether a three-cycle exists.

Below $1 + \sqrt 8$ it does not: three of the humps and dips of $f^3$ come close to the diagonal and miss it. As $r$ increases the graph moves, and at $1 + \sqrt 8$ it touches the diagonal at three points simultaneously — three points because a three-cycle is three points, and $f^3$ has the same graph shape near each of them. Past that parameter each touching point becomes two crossings, and the map has two three-cycles: one attracting, one repelling.

That event is a tangent or saddle-node bifurcation: a point that pulls and a point that pushes created together from nothing, which is the commonest way for fixed points to appear or disappear. The exact value $1 + \sqrt 8$ comes from algebra: dividing $f^3(x) - x$ by $f(x) - x$ leaves a polynomial of degree six whose roots are the three-cycles, and the parameter at which it first has real roots is where its discriminant vanishes, a condition that reduces to $r^2 - 2r - 7 = 0$.

Why three channels open together #

The graph of $f^3$ touches the diagonal at three places at the same parameter, and that simultaneity is not a coincidence of this map’s shape. It is forced.

Suppose $f^3$ is tangent to the diagonal at a point $p$, so $f^3(p) = p$ and the slope of $f^3$ there is exactly one. Then $q = f(p)$ also satisfies $f^3(q) = f(f^3(p)) = q$, and by the chain rule the slope of $f^3$ at $q$ is the product $f'(q)\,f'(f(q))\,f'(f^2(q))$ — the same three factors, in rotated order, as the slope at $p$. So the slope at $q$ is one as well, and $q$ is also a point of tangency. The same holds for $f(q)$. A tangency at one point of a would-be three-cycle is a tangency at all three, because the three points are the same cycle and the slope of $f^3$ is a property of the cycle rather than of the point.

The consequence for the time series is that the orbit has three channels to pass through, one after another, on each lap of the ghost cycle. They are the same channel seen at three places, since $f$ carries each onto the next, and the orbit spends the same number of laps in each. That is why a quiet stretch in the time series looks like a clean period-three pattern rather than a slow drift at one place: the orbit is creeping through all three channels at once, one step of $f$ per channel.

The channel the ghost leaves behind #

Just below the tangency the graph of $f^3$ misses the diagonal by a very small amount. Near each of the three almost-touching points there is a narrow passage between the curve and the line, and an orbit that enters the passage has to go through it.

The staircase is the cobweb construction applied to $f^3$: go up to the curve, across to the diagonal, up to the curve again. Where the curve is far from the diagonal the steps are large and the orbit moves quickly. In the channel the curve is barely above the line, the steps shrink to almost nothing, and the orbit creeps. Each creeping step of $f^3$ is three steps of $f$ that land almost where they started — which is exactly what the shaded stretches of the time series recorded.

The cycle that will exist at $1 + \sqrt 8$ is not there yet, but the geometry that will make it is. The orbit slows down where the fixed point is about to be born, as though it were attracted to a point that does not exist. That is sometimes called the cycle’s ghost, and the quiet stretches are the time spent near it.

Why the time grows like one over a square root #

The passage time can be computed, and the computation needs nothing about the logistic map except that the curve is tangent to the line at the critical parameter.

Near a point of tangency, write $y$ for the distance along the diagonal from that point and $\varepsilon$ for how far below the critical parameter $r$ is. To leading order the third iterate looks like

$$y_{n+1} = y_n + a y_n^2 + b\,\varepsilon,$$

for some positive constants $a$ and $b$: the $a y^2$ term is the curve bending away from the line, which it does quadratically because it is tangent there, and $b\varepsilon$ is how far the curve has lifted clear. When both are small each step changes $y$ by very little, and the sequence behaves like the solution of the differential equation

$$\frac{dy}{dn} = a y^2 + b\varepsilon.$$

That equation integrates to an arctangent: $y = \sqrt{b\varepsilon/a}\,\tan\!\big(\sqrt{ab\varepsilon}\,n\big)$. The orbit passes from large negative $y$ to large positive $y$ — through the whole channel — as the tangent passes from $-\infty$ to $+\infty$, which takes $n = \pi/\sqrt{ab\varepsilon}$ steps. So the passage time grows as $1/\sqrt{\varepsilon}$: a hundred times closer to the window, the orbit takes ten times as long to get through.

The derivation uses nothing but the local shape of a tangency. That is the whole reason the law is universal — every tangent bifurcation of every smooth map has a channel of this shape, and every one produces passage times with the same square root.

The law, measured #

The measured slope is $-0.53$, against the predicted $-1/2$. The small excess is expected and the direction is instructive. The derivation computes the time to cross the whole channel, but an orbit is reinjected into the channel by the chaotic burst that preceded it, and it lands at a random position — sometimes near the entrance, sometimes halfway through. The mean quiet phase is the passage time averaged over those entry points, and at the larger distances, where the channel is short, a larger fraction of the entries are part-way along. Closer to the tangency the channel is long compared with the spread of entry points, and the slope approaches a half.

The quiet phases are also defined by a threshold — each value within a hundredth of the value three steps before, for more than twenty steps — and a different threshold shifts every mean by a similar factor without changing the slope. The slope is the measurement; the individual means are partly conventions.

Stretches, then bursts #

Two features of the longer time series are worth pointing out, because they are what makes intermittency recognisable in data that did not come from a known equation.

The bursts do not lengthen as the window approaches. They are excursions of the orbit through the rest of the interval, where the map is as chaotic as it ever was, and their duration depends on how long the orbit takes to find its way back to a channel entrance — a property of the chaotic region, not of the tangency. Only the quiet phases grow. So the fraction of time the orbit spends looking periodic rises toward one, and in the limit the orbit is almost always quiet and almost never predictable about when it will stop.

The quiet phases vary a great deal individually. The short ones in the figure are reinjections near the exit; the long ones are reinjections near the entrance. The distribution of phase lengths is set by the distribution of reinjection points, which depends on the global dynamics, and it is not universal even though the mean’s scaling is.

Chaos that fades as a square root #

The square-root law has a second consequence, and it concerns how chaotic the orbit is overall rather than how its time is divided.

The rate at which nearby orbits separate is the Lyapunov exponent: the long-run average of $\log|f'|$ along the orbit. During a burst the orbit is in the chaotic part of the interval and the average of $\log|f'|$ is positive, much as it is anywhere else in the grey region. During a quiet phase the orbit creeps through channels where $f^3$ has slope very close to one, so over each lap of three steps the logarithms of the slopes add to nearly nought, and the time spent there contributes almost nothing to the average.

So the exponent is roughly the bursts’ contribution divided by the total time, and the total time is dominated by the quiet phases, whose lengths grow as $1/\sqrt\varepsilon$. The exponent therefore falls toward nought like $\sqrt\varepsilon$ as the window approaches — the orbit remains chaotic right up to the edge, and becomes less chaotic in a precisely measured way, before dropping discontinuously to a negative value when the attracting cycle is born. The Lyapunov curve across the diagram shows this as a sharp dip into the window with a rounded shoulder on its left side, and the shoulder’s shape is the square root.

That is also the honest answer to a question the bifurcation diagram cannot answer by itself: is the orbit just below the window really chaotic, or merely slow to settle? It is really chaotic, with a positive exponent at every parameter below $1 + \sqrt 8$; the exponent is simply small, and a finite computation of it at a parameter close to the edge will be dominated by whichever phase it happens to end in.

The same square root at the edge of a lock #

The derivation above assumed a tangency and nothing else, and a tangency has already been drawn in a completely different setting.

When a periodically forced oscillator locks to its forcing, the locking holds over a plateau of parameters — the Arnold tongues of the circle map — and at the plateau’s edge the lock comes apart by exactly this kind of collision: a stable and an unstable periodic orbit meeting and vanishing. Just outside the plateau the oscillator is not locked, but it behaves as though it nearly were. Its relative angle drifts slowly for a long time, then slips by a full turn quickly, then drifts slowly again.

Those phase slips are intermittency, and the time between slips grows as one over the square root of the distance from the plateau’s edge, for the reason derived above. It is the same law that governs a clock pendulum drifting out of sync with a driving signal, a laser just outside its injection-locking range, and — in the model that gave the phenomenon its name — the onset of turbulence in convecting fluids, where Yves Pomeau and Paul Manneville described three types of intermittency in 1980 and this one became type I.

That is the connection worth carrying out of the logistic map. A window opening in a chaotic interval map and a lock breaking in an oscillator look like unrelated events. They are the same bifurcation, and anybody measuring the lengths of quiet stretches in either can read off the square root.

What the window does next #

The tangency creates two three-cycles, and they have opposite fates that between them shape the rest of the window.

The attracting cycle is the one the diagram shows. As $r$ increases its multiplier falls from one through nought — the superstable point, near $3.8319$ — and on to $-1$, where it period-doubles to six, then twelve, in a cascade of its own whose gaps shrink by the same Feigenbaum ratio. Beyond that cascade the window fills with three bands of chaos, each a small copy of the whole diagram’s chaotic region.

The repelling cycle is invisible in the diagram and ends the window. As the three chaotic bands widen, they eventually reach the repelling cycle’s points, which sit just outside them. The moment a band touches a repelling point, the orbit can escape along it into the rest of the interval, and the three narrow bands abruptly become one wide band. That event, near $r \approx 3.8568$, is an interior crisis, and just beyond it the orbit shows a second kind of intermittency: long stretches confined to the old three bands, broken by bursts through the region that has just opened.

So the saddle-node bifurcation that opens the window produces, in the same instant, both the cycle that fills the window and the unstable twin that will close it. The window’s life is the interval between the twins’ birth and the moment the chaos grown from one of them collides with the other.

What the time series cannot settle #

The quiet phases are identified by a threshold, and a threshold is a choice. Nothing in the dynamics marks where a quiet stretch begins; the orbit’s closeness to a period-three pattern changes continuously, and a different threshold would start and end every quiet stretch at slightly different steps. The figures report means and a slope, both of which are robust to the choice; the individual phase lengths printed in the captions are not.

The square-root law is derived for the idealised map $y \mapsto y + ay^2 + b\varepsilon$ and checked against the logistic map at five parameters spanning two decades of $\varepsilon$. Two decades is enough to distinguish a slope of a half from a slope of one, and not enough to distinguish $-0.50$ from $-0.53$ on the data alone. The reason given for the excess is an argument about reinjection, not a measurement.

And the time series are floating-point orbits. At a hundred-thousandth below the tangency the channel is narrow enough that rounding error in each step is small compared with the gap, and the orbit’s passage is faithful. Much closer, the channel’s width becomes comparable with the arithmetic’s resolution and a computed orbit could be trapped or released by rounding rather than by the map.

Still open: where the other windows open #

The period-three window opens at an algebraic number with a two-line derivation, $1 + \sqrt 8$. Every other window also opens at a tangent bifurcation, so every one of their left edges is an algebraic number: the parameter where some polynomial’s discriminant vanishes. For period four the relevant polynomial is already of high degree, and its root near $3.9601$ was pinned down exactly only with computer algebra. For period five and beyond the polynomials grow so fast that exact expressions are impractical, and the onset parameters are known only numerically.

No pattern in those algebraic numbers is known, and none is expected from their construction. What is known about the windows as a family — that they are dense, that they are ordered by the kneading sequences of the critical orbit, that every one of them opens with a stutter of exactly this kind — comes from the topology of the map rather than from its algebra. The exact position of the edge of a particular window, beyond the first few, is a number that can be computed to any precision and has no known description.

Almost a cycle, for longer and longer #

The period-three window’s edge is sharp in the bifurcation diagram because the diagram only records where orbits settle. The time series shows the edge is soft in time: the approaching cycle makes itself felt long before it exists, as a channel the orbit has to squeeze through, and the squeeze lasts longer the closer the parameter comes.

That softening is general and precisely quantified. Wherever two fixed points are about to be created from nothing, the system nearby spends a time proportional to one over the square root of its distance from the event behaving as though they already were — a stutter whose length is a measurement of how close the collision is, in a logistic map, a forced pendulum or a fluid on the verge of turbulence.