A whole interval of speeds

The circle map advances a point round a circle by a fixed amount $\Omega$, nudged back and forth by a sine of strength $K$:

$$x \mapsto x + \Omega - \frac{K}{2\pi}\sin 2\pi x \pmod 1.$$

Every essay on it so far has stayed at or below $K = 1$, and for a good reason. There, the staircase that is flat almost everywhere holds: every orbit has the same average advance per step, the rotation number, and it depends only on $\Omega$ and $K$. How a lock comes apart explains the plateaus of that staircase as collisions of periodic orbits and notes, in passing, that at $K = 1$ the plateaus fill the axis and that above it they would have to overlap — which for an ordinary circle map is impossible, since its rotation number is a single number.

Above $K = 1$ the map is no longer ordinary, and the overlap happens. This essay is about what an overlap of rotation numbers looks like: one map with many average speeds at once.

The fold #

The derivative of the map is $1 - K\cos 2\pi x$. For $K \le 1$ it is never negative, so the map only ever moves points forward in order: if $x < y$ then their images keep that order. For $K > 1$ the derivative goes negative near $x = 0$, and the map turns back on itself there.

The figure draws the lift — the map as a function on the real line, recording how far a point has gone round rather than only where it lands. At $K = 0.6$ the lift rises steadily. At $K = 1.8$ it rises, falls and rises again, so two different points near the fold land on the same image. The map has stopped being a one-to-one correspondence of the circle with itself; it can no longer be run backwards, and the theory of the rotation number — which begins from the fact that an order-preserving map carries arcs to arcs — no longer applies.

What still makes sense is the average advance of each individual orbit, $\lim (F^n(x) - x)/n$ when the limit exists. The question is how many different values it takes.

Two rising maps that bracket the folded one #

The answer comes from a construction that repairs the fold in two opposite ways. Above the lift, take the smallest rising function that is never below it: at each $x$, the largest value the lift reaches anywhere to the left, $F_+(x) = \max_{y \le x} F(y)$. It follows the lift where the lift rises and runs flat across the fold. Below the lift, take the largest rising function never above it: $F_-(x) = \min_{y \ge x} F(y)$, which runs flat across the dip from the other side. Both are drawn dashed in the figure.

Each envelope is a non-decreasing map of the circle’s lift — an ordinary circle map, with an ordinary rotation number — and between them they bound every orbit. If an orbit of $F$ starts at $x$, then after $n$ steps it is at most $F_+^n(x)$ and at least $F_-^n(x)$, because each envelope sandwiches the lift and each preserves order. So every orbit’s average advance lies between the rotation number of $F_-$ and that of $F_+$:

$$\rho_- \le \lim \frac{F^n(x) - x}{n} \le \rho_+.$$

The theorem that makes this more than a bound is that every number in between is achieved. Work of Ryuichi Ito, of Sheldon Newhouse, Jacob Palis and Floris Takens, and of Michał Misiurewicz, all in the early 1980s, established that the set of average advances of a degree-one circle map is exactly the closed interval $[\rho_-, \rho_+]$ — the rotation interval — and that every rational number $p/q$ inside it is the average advance of a periodic orbit that goes round $p$ times in $q$ steps.

Two staircases #

At a fixed $K$ above the critical line, each end of the interval is a function of the drive, and each is a staircase of its own.

The upper and lower ends climb from nought to one as $\Omega$ runs across, each with plateaus at the rationals, and the shaded band between them is where one map has many speeds. At $K = 1.5$ the band is widest near $\Omega = 0.43$, from $0.4$ to $0.5$ — every fraction between two fifths and a half is the speed of some orbit of that one map. Elsewhere the band pinches shut on the widest plateaus, at nought, a half and one, where both ends lock onto the same fraction and the map still has a single speed.

The two staircases are exactly the plateaus of the circle map overlapping, as the essay on how locks come apart anticipated. Below $K = 1$ the plateaus of different fractions sit side by side, each drive belonging to at most one, and the two ends coincide. Above it the plateau of $2/5$ and the plateau of $1/2$ both claim the drives near $0.43$, and a map in both claims both speeds — and every speed between.

Where the interval opens #

Sweeping both the drive and the strength shows where in the plane of parameters the map has more than one speed.

Below the dashed line at $K = 1$ every cell is pale: one speed. Above it the picture is organised by the tongues — the wedges of parameter space in which a given fraction locks. The widest tongues, for rotation numbers nought, a half and one, keep a single speed well above the line, because inside them one attracting periodic orbit and its fraction dominate both envelopes. Between them the neighbouring tongues overlap and the interval opens, first narrowly and then, by $K = 2.5$, across most of the drive, wide enough in places to hold every fraction from nought to a half.

At the resolution drawn, a grid of sixty drives by forty strengths, the opening appears to start somewhat above the line rather than at it. That is a limit of the grid, not a fact about the map: the regions where the interval first opens just above $K = 1$ are thin wedges between tongues, narrower than a grid cell, and they are there however close to the line one looks.

A periodic orbit for every fraction inside #

The theorem’s second half — every fraction in the interval is realised by a periodic orbit — can be checked orbit by orbit, because a periodic orbit of speed $p/q$ is a solution of an equation: a point $x$ that the lift carries to exactly $x + p$ in $q$ steps.

At $\Omega = 0.3$ and $K = 2$ the interval runs from $0$ to $1/3$. The fractions inside it with denominator at most six are $0, 1/6, 1/5, 1/4$ and $1/3$, and for each the equation has a solution: a periodic orbit going round once in six steps, once in five, once in four, once in three, and a fixed point that does not go round at all. The eight fractions outside the interval with the same denominators, from $2/5$ to $5/6$, have no solution anywhere on the circle — the search over the whole circle finds nothing, as the theorem requires. The interval is not a range of approximate speeds; it is an exact list of which speeds exist.

The last column is the stability of each orbit, its multiplier: the product of the map’s derivative round the orbit. Below one in size the orbit attracts its neighbours; above one it repels them. The fixed point has multiplier $0.33$ and attracts. Every other orbit in the list repels, with multipliers from about $5$ to $83$ — the orbits exist, but anything that starts near one of them is pushed away.

What a random starting point sees #

That has a consequence that is easy to miss and important in practice.

A thousand orbits started at random all settle on the same speed, nought: every one is drawn into the attracting fixed point. The interval from nought to a third, the speeds of five periodic orbits in the table and infinitely many others, is invisible. Nothing in the long-run behaviour of a typical orbit reveals that the map has any other speed.

This is the difference between the rotation interval, which lists the speeds of all orbits, and what an experiment measures, which is the speed of the orbits a random start reaches. Above the critical line the two come apart. The interval is still the right object for the mathematics — its width forces chaos, since a map with two different speeds must have orbits that switch between them in every possible pattern — but that chaos lives on a set of starting points too thin to hit by chance, while the attracting orbit takes everything else. In other parameter ranges the attractor is itself chaotic, and then a random orbit does wander across part of the interval; but even then it samples the interval in its own proportions rather than exhibiting all of it.

Why two speeds force every pattern #

The claim that a wide interval forces chaos has a short argument, and it uses nothing but two periodic orbits with different speeds.

Take the fixed point, which does not go round, and the orbit of speed $1/3$, which goes round once every three steps. Near each is a small arc that the map, applied the right number of times, stretches across both arcs — the fold makes the stretching possible, since a folded map can carry an arc forward and bring part of its image back. So from any arc a point can be chosen to follow, at the next stage, either orbit’s route, and then again, and again: every infinite sequence of choices, stay or go round, is followed by some starting point. That is an orbit written as a word, with every word allowed, and the number of distinct patterns of length $n$ grows exponentially — positive topological entropy, the measure of chaos that counts a map’s folds. Each such orbit’s average speed is whatever fraction of its steps went round, which is how every speed between nought and a third is reached.

Below the critical line none of this can happen, because an order-preserving map of the circle carries arcs to arcs without bringing any back, and a map that cannot fold cannot stretch an arc over two different places. The fold is the whole mechanism: it is what makes a map stretch and fold in the way that manufactures chaos, and the rotation interval is the bookkeeping of what the folding has done to the speeds.

The same fold, one dimension down #

A folded circle map is a relative of the most familiar chaotic map, the logistic map, which folds the unit interval over itself once. The logistic map has no rotation number — it does not go round anything — and its route to chaos, the road paved with doublings, is a cascade of periods $1, 2, 4, 8, \dots$ rather than a staircase of fractions. The circle map above $K = 1$ has both kinds of structure at once. Inside each tongue it behaves like a logistic map, and its locked periodic orbit doubles its period as $K$ rises, exactly as the logistic map’s does; between tongues, where the interval is open, it has the fractions as well.

That is the sense in which the circle map sits between the two classic pictures of the subject. At $K = 0$ it is a rigid rotation, with its three gaps; up to $K = 1$ it is a rotation with plateaus; above $K = 1$ it folds, and every behaviour of the logistic map appears inside each plateau while the plateaus themselves overlap. The fixed point that attracted every random start in the last figure is the first step of such a cascade: raise $K$ further at the same drive and it doubles, and doubles again, before the attractor becomes chaotic.

What a physical pendulum shows #

The circle map began as a model of a driven oscillator — a pendulum pushed periodically, an electric circuit driven at one frequency and oscillating at another, the phase of a heart cell stimulated by a pacemaker — and the region above the critical line is where those systems become hard to predict.

In a driven pendulum, strong driving is the regime of above $K = 1$. There the pendulum can settle into different patterns of rotation depending on how it was started, two patterns with different average speeds coexisting for the same drive: that is hysteresis, the same drive producing different behaviour depending on the history, and it is the rotation interval made visible — the attracting orbits of two different fractions inside one interval. In superconducting junctions driven by microwaves, the steps in voltage named after Sidney Shapiro are the plateaus of the staircase, measured in a laboratory, and strong driving brings the same coexistence of speeds.

The mathematics adds one thing the experiment cannot. However the pendulum is started, it shows one speed at a time, the speed of whatever attracts from that start. The interval says what else is possible: every fraction in it is a genuine periodic motion of the system, most of them unstable and therefore never seen, and between them a set of chaotic motions that switch among speeds without settling.

What the pictures cannot show #

That the interval is exactly the set of speeds. The staircase figure computes each end as the rotation number of an envelope map, over a finite run of a sampled function; the theorem that every speed in between is realised, and none outside, is Ito’s and Misiurewicz’s, and the periodic-orbit table checks it for five fractions at one parameter.

The chaos. A non-degenerate interval forces positive topological entropy — exponentially many different orbit patterns — and none of the figures shows an orbit switching speeds. The histogram shows the opposite, a single attracting speed, because the switching orbits are a set of starting points of no length.

Which orbits the table found. For each fraction the search reports one point of one periodic orbit. There may be several orbits with the same speed — typically an attracting and a repelling one, or two repelling — and the table does not say how many; it says only that at least one exists.

The thin regions. The parameter-plane figure has cells of width one sixtieth by one sixteenth; the fine structure of where the interval first opens just above $K = 1$ is below its resolution.

Still open: how the interval’s ends move #

Each end of the rotation interval is a monotone function of the drive with a plateau at every fraction — two devil’s staircases. For a fixed $K$ between one and the point where the envelopes stop being well behaved, the structure of these staircases is understood in outline: plateaus at every rational, filling the drive axis. What is not understood is how the two staircases are related, and in particular how the width of the interval behaves as $K$ decreases towards one — how fast the band between the two staircases closes as the fold flattens out.

If the renormalisation picture that explains the critical line extends above it, the width should obey scaling laws with universal exponents, like those found at $K = 1$ exactly, where the unlocked parameters form a set of dimension about 0.87. Whether it does is a question on which numerical studies exist and a proof for the circle map itself does not — the same situation as the renormalisation theory at the critical line, extended into the region where the map folds.

A second question is simpler to state. For $K$ well above one the envelopes are flat over long stretches and the interval is wide, and it is natural to ask for the drive at which the interval is widest and what its width is, as a function of $K$ — in effect, the shape of the shaded band in the parameter-plane figure. The answer is known numerically for particular values and is not known in closed form for any $K$ above the line: the band’s boundary is assembled from the edges of infinitely many overlapping tongues, and no formula for it has been found.