Two chaotic orbits made to agree

The defining experience of chaos is two orbits that part. Start the map $x \mapsto 4x(1 - x)$ at $0.3$ and at $0.3001$, and within a few dozen steps the two sequences have nothing in common; the difference of one part in ten thousand has doubled at every step, on average, until it is as large as the whole interval. The doubling rate — a factor of $2$ per step, a Lyapunov exponent of $\ln 2$ — is the number that makes the map unpredictable.

Now let the two copies see each other. At every step each copy computes its own next value, and then each takes a fraction $\varepsilon$ of the other’s next value in place of the same fraction of its own:

$$x' = (1 - \varepsilon)\,f(x) + \varepsilon\,f(y), \qquad y' = (1 - \varepsilon)\,f(y) + \varepsilon\,f(x).$$

With $\varepsilon = 0$ they are independent and chaotic. With $\varepsilon = \tfrac12$ both copies take the average of the two next values and become identical after a single step. Somewhere between the two the behaviour must switch, and the question this essay answers is where — and why the answer is a formula in the exponent that measured the chaos.

What the coupling does to a small gap #

The hero figure shows the switch in action. At $\varepsilon = 0.15$ the two copies behave like two independent chaotic orbits: the gap grows to the size of the interval within twenty steps and stays there, wandering. At $\varepsilon = 0.3$ the gap shrinks steadily, about twenty per cent a step, falls below the precision of the arithmetic after some seventy steps, and stays at nought for good.

The second picture should be read carefully, because it is not two orbits settling down. Each copy is still running the chaotic map. Its values still jump around the interval, never repeating, exactly as unpredictable as before. What has stopped is the disagreement: the two copies are now running the same chaotic orbit, together.

Why they lock is a three-line calculation. Write $d = x - y$ for the gap. Subtracting the two coupling equations,

$$d' = (1 - 2\varepsilon)\,\big(f(x) - f(y)\big),$$

and when the gap is small, $f(x) - f(y)$ is approximately $f'(x)\,d$: the map stretches a small gap by its slope, just as it does for two uncoupled orbits. So each step multiplies the gap by $(1 - 2\varepsilon)\,f'(x)$. Over many steps the slope’s contribution averages out, in logarithmic terms, to the Lyapunov exponent $\lambda$, and the coupling contributes a steady factor of $|1 - 2\varepsilon|$. The gap’s own exponent — the transverse exponent, measuring growth across the line where the copies agree — is

$$\lambda_\perp = \lambda + \ln|1 - 2\varepsilon|.$$

The copies lock when it is negative. For $f(x) = 4x(1 - x)$, with $\lambda = \ln 2$, that means $2\,|1 - 2\varepsilon| < 1$, or $\tfrac14 < \varepsilon < \tfrac34$. At $\varepsilon = 0.3$ the gap shrinks by $2 \times 0.4 = 0.8$ per step, as the figure shows; at $0.15$ it grows by $2 \times 0.7 = 1.4$.

Agreeing to every digit, in arithmetic and in fact #

The figure’s gap reaches nought, and it is worth being exact about what that means. In the arithmetic of a computer, once the difference between the two copies falls below the last digit a double can carry, the two values are identical, and from then on the copies compute exactly the same thing at every step: the gap is nought for ever, not merely small. In exact arithmetic the gap never reaches nought. It shrinks by a factor of $0.8$ per step for ever and is always positive.

Both descriptions are faithful to the mathematics in the sense that matters. The computed orbit of a chaotic map is not the true orbit of its starting point — after a few dozen steps the rounding errors have grown to the size of the interval — but, by the shadowing property that the orbit a computer draws examined, it stays close to the true orbit of some other starting point. The locked copies share that shadow: the computer has replaced two orbits converging exponentially fast with one orbit, and the replacement is accurate to the precision of the arithmetic. The figure’s claim is the transverse exponent, and the exponent is the same in either arithmetic.

The exponent that measured the chaos sets the price #

The formula says something that is worth stating as a principle: the more chaotic the system, the harder it is to lock, and “more chaotic” means exactly “larger Lyapunov exponent”. The figure below tests the formula for two maps of different chaos.

The dots are measured by following a long orbit and averaging the logarithm of the factor $(1 - 2\varepsilon) f'(x)$ at each step; the curves are the formula. They agree to the figure’s precision everywhere except within a sliver of $\varepsilon = \tfrac12$, where the factor is nought and the logarithm is minus infinity — where the copies lock in one step. For the less chaotic map at $r = 3.8$ the window is wider: its exponent of $0.432$ needs only $|1 - 2\varepsilon| < e^{-0.432} = 0.649$, so a coupling of $0.175$ suffices.

The threshold therefore has a clean general form, $\varepsilon_c = (1 - e^{-\lambda})/2$, which converts an exponent into a price. A system whose errors double every step needs a quarter of each copy’s next value replaced by the other’s. A system whose errors grow by half as much per step, in logarithmic terms, needs less. The quantity that told how many steps of prediction a map allows also tells how hard it must be pulled to agree with a copy of itself.

Following instead of meeting halfway #

The coupling so far has been mutual: each copy pulls on the other. If instead one copy runs undisturbed and the other is pulled towards it — a drive and a response — the arithmetic changes in an instructive way.

With one-way coupling the follower obeys $y' = (1 - \varepsilon) f(y) + \varepsilon f(x)$ while $x' = f(x)$, and the gap is multiplied by $(1 - \varepsilon) f'$ per step rather than $(1 - 2\varepsilon) f'$. The transverse exponent is $\lambda + \ln(1 - \varepsilon)$, and at $\lambda = \ln 2$ it is negative only for $\varepsilon > \tfrac12$. A follower needs exactly twice the pull, because the leader does none of the closing.

Louis Pecora and Thomas Carroll made this the basis of their 1990 paper on synchronised chaos, which showed that a chaotic electronic circuit could drive a second one into perfect agreement through a single transmitted signal. The application they and others proposed was secure communication: a message hidden in a chaotic signal, readable only by a receiver built to synchronise with the transmitter, since anyone without an identical circuit sees only chaos. The scheme turned out to be breakable — the chaos used was low-dimensional enough to be reconstructed from the signal — but the phenomenon it rested on is now routine in laser arrays, electronic circuits and models of neural populations.

Locked on average, and bursting apart anyway #

The threshold formula uses the average stretching. Chaotic orbits do not stretch evenly: they spend stretches of time near points where the slope is unusually large, and while they are there, the local transverse factor can exceed one even when the average is below it.

At $\varepsilon = 0.18$ the copies are locked on average: the transverse exponent is slightly negative, so a perturbation dies away over the long run. But the noise keeps supplying fresh small perturbations, and whenever the orbit wanders into a region where the local stretching beats the coupling, the gap is amplified, sometimes all the way up to the size of the interval, before the average contraction brings it back. The record is long quiet stretches near the noise level, broken by sudden bursts — the pattern called on-off intermittency, described by Nathan Platt, Edward Spiegel and Cyrus Tresser in 1993.

The worst offender for this map is its fixed point at $x = 1 - 1/3.8 \approx 0.737$, where the slope is $-1.8$, steeper than the average stretching factor of $e^{0.432} \approx 1.54$. Holding the copies together at the fixed point would need a coupling of $0.222$, more than the average threshold of $0.175$. Between the two values, the lock is real on average and fragile in practice, and any noise at all will occasionally find the weak spot. The phenomenon was named bubbling by Peter Ashwin, Jorge Buescu and Ian Stewart in 1994, and it is the same distinction that the damped random Fibonacci sequence drew between the typical rate and the rare runs that dominate an average: a threshold set by an average says nothing about what the worst stretches do.

The surprising thing: rings that cannot lock #

Coupling two copies is the start of a larger question: whether a whole population of chaotic units can be brought into step. The simplest population is a ring, each unit pulled equally by its two neighbours.

The analysis follows the two-copy calculation with one change. A disagreement among $N$ units can take many shapes — one unit out of step, alternate units out of step, a slow wave of disagreement running once round the ring — and each shape is shrunk by its own factor per step, $|1 - \varepsilon(1 - \cos 2\pi k/N)|$ for the $k$-th wave. The fastest pattern, alternating units, is shrunk by $|1 - 2\varepsilon|$, the same factor as for a pair; the slowest, one wave round the ring, by $|1 - \varepsilon(1 - \cos 2\pi/N)|$, which is close to one in a large ring because neighbouring units in a slow wave hardly disagree. Every pattern must shrink faster than the chaos stretches it, by a factor of two for this map.

That turns into a window for $\varepsilon$, and the window closes at six. For the slowest wave to be shrunk enough, $\varepsilon$ must be large; for the fastest not to overshoot — a factor $|1 - 2\varepsilon|$ that has become larger than one half because $\varepsilon$ is near one — $\varepsilon$ must not be too large. In a ring of five the two requirements leave a narrow band, around $0.72$ to $0.83$. In a ring of six they contradict each other, and no coupling of this kind can synchronise the ring at all. The simulation at the centre of the five-ring’s window confirms both halves: five lock, six never do.

It is a strange result to meet without warning. Coupling more strongly is the obvious remedy for a population that will not lock, and in a ring it stops working at six units, because strength helps the slow patterns of disagreement and harms the fast ones. Louis Pecora and Thomas Carroll turned this into a general method in 1998 — the master stability function, which separates the unit’s chaos from the network’s shape — and it explains why chaotic oscillators coupled along a long chain or ring do not settle into unison by nearest-neighbour coupling alone, while the same oscillators coupled all to all can.

Huygens’ clocks, and the chaos that came after #

Synchronisation was first noticed in clocks, not chaos. In February 1665 Christiaan Huygens, ill in bed, observed that two pendulum clocks hanging from the same beam swung in exactly opposite phase, and that when he disturbed one they returned to that state within half an hour. The coupling was the beam’s tiny motion, and the clocks were periodic, which makes locking easy to understand: each periodic oscillator sits somewhere in its own cycle, the coupling nudges that position forward or back, and two positions nudged towards each other meet. The circle map’s staircase is the mathematics of that kind of locking, in which a whole interval of driving speeds produces the same rational ratio of rotations.

Chaotic systems have no phase to nudge. Their orbits never repeat, nearby orbits separate exponentially, and for a long time it seemed obvious that two of them could not be made to march together. Hirokazu Fujisaka and Tomoji Yamada showed in 1983 that they could, by exactly the calculation above, and the result was largely ignored until Pecora and Carroll’s circuits made it impossible to ignore. The surprise was not that coupling helps; it was that the needed strength is finite — that a system sensitive to every perturbation can be held to a copy of itself by a fixed fraction of its own next step.

Two systems forgetting their difference #

There is a sense in which synchronisation is the opposite of chaos and a sense in which it is the same thing seen from the side. Chaos is sensitivity to the starting point along the orbit; synchronisation is insensitivity to the starting difference across it. The random products of the previous essay showed both at once — random matrices forget the starting direction while growing at a definite rate — and coupled maps do the same: the common orbit is as sensitive as ever, while the difference between the copies is erased.

Probability has the same idea under the same name. Two copies of a random process run on shared random numbers — a coupling, in the language of mixing times — come together and stay together, and the time they take to meet bounds how fast the process forgets its start. Coupled chaotic maps are the deterministic version: the shared input is the other copy’s next value, and the transverse exponent plays the role the meeting time plays for a random walk.

A linear threshold, and the riddled basins it cannot see #

The threshold formula is linear. It describes what happens to a gap that is already small, and says nothing about gaps that start large. Two copies started far apart, with a coupling inside the window, usually lock anyway, but not always: for some maps and couplings the set of starting pairs that eventually lock is riddled — every pair that locks has, arbitrarily close to it, pairs that do not. Whether that happens for the maps drawn here is not something the figures test; they start the copies close together.

The bursts depend on the noise and on the run. The on-off figure uses noise of one part in a million and six thousand steps, and the fraction of time spent apart, about a tenth, would change with either. At smaller noise the bursts are rarer and the quiet stretches longer, and a short run could show no bursts at all, which is precisely why the phenomenon was missed in early experiments.

The ring result is for one map and one kind of coupling. The window that closes at six comes from the doubling factor of $x \mapsto 4x(1 - x)$ and from coupling each unit to its two neighbours equally. A less chaotic map leaves a wider window and allows longer rings to lock; coupling to more distant neighbours changes the shrink factors of the waves. The figure computes one case exactly and the general rule — that the slow waves of a large ring defeat local coupling — rather than a universal bound.

Still open: synchronisation in networks #

For identical units the master stability function settles whether a network of any shape can lock, by reducing the question to the network’s eigenvalues and one function of the unit. Real populations are rarely identical. Fireflies flash at slightly different natural rates, heart cells have different thresholds, and neurons differ in everything. For units that differ, the transition to synchrony is gradual rather than sharp, and in Yoshiki Kuramoto’s model of coupled oscillators — periodic rather than chaotic — it happens at a critical coupling that is known in the limit of many oscillators. For chaotic units that are only nearly identical, and for networks whose shape changes as they run, how much coupling is needed, and whether the locked state is stable against the bursts drawn above, are questions answered case by case.

And the ring result raises a question it cannot settle. Synchronisation in long chains of chaotic units fails under nearest-neighbour coupling, yet spatially extended systems — a turbulent fluid, a beating heart — sometimes do organise into coherent patterns. Which patterns a long chain of chaotic units settles into instead of unison, and how they depend on the coupling, is a large subject in its own right, of which these maps are the simplest example, and which is far from mapped.