The map a dripping tap turns out to be

There is a celebrated result about one-dimensional maps: turn a knob on any smooth map with a single hump and it period-doubles its way to chaos, with the gaps between successive doublings shrinking by 4.6692 whatever the map is. The mathematics of it — the constant, the renormalisation argument that produces it, the classification of what shares it — belongs to matpic.com, which derives it, and this collection takes it as given rather than deriving it again.

What that result does not say is why it should matter to anyone holding a dripping tap. A tap is a continuous body of water with surface tension and viscosity in it; a convection cell is a fluid with an unlimited number of degrees of freedom; neither is a one-dimensional map, and nothing about either suggests it should be. The physics question is how a system like that becomes one — and the answer is why the same number keeps being measured in laboratories that have no map anywhere in them.

Start with what the map does, because the rest of the essay is an argument that a laboratory is doing it.

The mechanism at each doubling is the same and it is worth stating once. A cycle is stable when a small displacement from it shrinks — when the derivative of the map, compounded round the cycle, has magnitude less than one. As the knob turns, that multiplier moves; when it passes through $-1$, displacements alternate in sign and grow, and the cycle stops being stable. But a displacement that alternates in sign returns to itself after two steps, so what emerges is a cycle of twice the period, and it is born stable. Turn the knob further and the same thing happens to it.

That is a local argument about one bifurcation, and it does not by itself predict a cascade. What makes a cascade is that the doubled cycle is a copy of the original problem: its second iterate, restricted to a small interval, looks like the original map with a different knob setting. So the argument repeats, and it repeats faster each time.

The whole cascade at once #

The gaps shrink geometrically, which is why an infinite sequence of doublings fits inside a finite interval of the knob. The accumulation point for this map is at $r = 3.569946$, and past it the orbit no longer visits a finite list of values at all.

Locating the doublings precisely is harder than it looks, and the way round the difficulty is worth knowing because it is the reason the numbers here can be computed rather than quoted. At a bifurcation the cycle’s multiplier is exactly $-1$, so the cycle is neutrally stable and iterating converges arbitrarily slowly — the thing to be measured is the thing that makes measurement fail. The fix is to locate the superstable settings instead: the knob values at which the cycle happens to pass through the map’s own turning point, where the derivative is zero and the multiplier with it. Those can be found by bisection to machine precision, they interleave the bifurcations, and the ratios of the gaps between them converge to the same limit.

The number, taken as given #

The figure computes the ratio twice, from two maps with nothing algebraically in common, because a number this essay is going to compare against laboratory measurements should be computed here rather than copied. It comes out at $\delta = 4.6692016$ both times.

The convergence is slow, and that is the fact the experimental sections need. The early ratios are 4.75, 4.66, 4.67 — the sequence is within a per cent by the fourth doubling and not before — so an experiment that reaches four doublings and reports 4.4 has not disagreed with anything. Reading the figure’s own early points against the limit is the honest way to say how many doublings a measurement needs, and the answer is more than any experiment has managed.

The experiments that had no map in them #

A constant computed from a one-line iteration would be a curiosity if it stayed there. It does not, and the reason is that a great many physical systems, sampled once per drive cycle, are a one-dimensional map with a smooth maximum.

A dissipative system driven periodically forgets its initial condition, so its long-run behaviour lives on a low-dimensional attractor; if the attractor is thin enough, the map from one drive cycle to the next is effectively one-dimensional. Where that holds, the cascade must appear and the ratio must be 4.669 — and it is a prediction about an apparatus whose equations nobody has written down.

The measurements are worth listing with their numbers, because what they had in common was the reduction and nothing else.

Liquid helium in a convection cell, 1979: a box a few millimetres across, heated from below, with the temperature at one point recorded against time. Four doublings were resolved before the noise floor, and the gap ratios came out near 4.4. That is not a disagreement — the figure above shows the ratio sequence still at 4.66 by the fourth term and approaching from either side — and it was the first physical measurement of the cascade in a fluid.

A driven nonlinear circuit, 1981: a diode and an inductor, driven by a signal generator, with the drive amplitude as the knob. It doubled four or five times and returned δ = 4.5 ± 0.6, and the appeal of it is that it costs nothing and shows the cascade on an oscilloscope in an afternoon.

A dripping tap, from about 1984: the knob is the flow rate and the observable is the interval between drops. The intervals settle to one value, then alternate between two, then four. Nothing about a pendant drop is one-dimensional — it is a fluid with surface tension, viscosity and inertia — and the return map reconstructed from the intervals is a hump.

A forced pendulum, in many laboratories since: the most direct mechanical case, and the one the next section follows.

The practical value of this is a prediction that costs nothing. If a system is seen to double once and then again, the third doubling can be predicted before it is measured — it will occur about 4.67 times closer to the second than the second was to the first — and the accumulation point can be extrapolated from three doublings to within a per cent. That is a quantitative forecast about a system whose equations may be entirely unknown, made from two observations and a number that belongs to no system at all.

Why a continuous system has a map at all #

The reduction is the physics, and it is worth doing carefully rather than asserting.

A driven dissipative system lives in a phase space of many dimensions — for a convection cell, in principle infinitely many. Dissipation contracts volumes in that space at a rate set by the damping: a small blob of initial conditions shrinks, and it shrinks by different factors in different directions. After a while the blob is a thin sheet, then a thin ribbon, and the long-run motion lies on whatever is left.

Now sample once per drive cycle. The state at one sample determines the state at the next, so the sampled dynamics is a map — a Poincaré return map — of the attractor to itself, and the attractor’s dimension is the map’s. If the contraction is strong enough in all but one direction, that dimension is barely above one, and a single coordinate along the attractor is nearly enough to say where the system is.

That is the whole of the correspondence and every word of it is a statement about damping. The strength of the contraction decides whether the map is one-dimensional; the smoothness of the flow decides whether it has a smooth maximum; and the folding that any bounded stretching motion must do — the fold that has to be there — decides that it has a maximum at all rather than being monotonic.

There is a measurable version of the condition. The attractor’s dimension can be estimated from data, and where it comes out below about 1.2 the one-dimensional treatment holds and the constants should be the universal ones; where it comes out at 1.5 or more it does not. Libchaber’s helium cell measured a dimension close to one; a weakly damped oscillator driven hard does not, and its cascade departs from the prediction in exactly the way the reduction says it should.

The mechanical instance #

The map is a caricature and the systems it describes are not, so it is worth following one real oscillator into the cascade.

Take a pendulum with friction, driven by a torque at a fixed frequency. At small drive it settles into a motion repeating once per drive cycle. Raise the drive and the restoring torque’s departure from proportionality begins to matter — the same departure that makes an oscillator answer at frequencies it was not driven at — and at some amplitude the once-per-cycle motion loses stability to one that repeats every two cycles: the pendulum swings a little further on odd cycles than on even ones. Raise it again and the pattern takes four cycles to repeat, then eight.

Sampling that pendulum once per drive cycle turns it into a map. The state at one sample determines the state at the next, and because the friction contracts phase-space volume the states after a while lie on a thin sheet, so one coordinate along that sheet is nearly enough. That is the correspondence, and everything about it is approximate except the conclusion: where the sampled dynamics really is a one-dimensional map with a smooth maximum, the constant has to be 4.669.

The nonlinearity that produces the maximum in the pendulum’s case is the one that has been in this collection since the small-angle approximation was named as a lie. A pendulum’s restoring torque falls below proportionality at large angle, so its frequency depends on how far it is swinging, so a driven one has two possible amplitudes over a range of drive frequencies — and the boundary between them, followed as the drive rises, is where the doubling starts. The route to chaos in a pendulum begins in the same term as the leaning resonance and in the sensitivity of the period to the swing.

What the mathematics supplies, in one paragraph #

The argument that produces the constant is a renormalisation: the second iterate of the map, restricted to the small interval the two-cycle keeps returning to and rescaled, is another map of the same family, so there is an operation taking maps to maps. That operation has a fixed point, the fixed point does not remember which map the operation started from, and the two constants — $\delta = 4.6692$ for the knob and $\alpha = 2.5029$ for the values — are eigenvalues of it. None of that is a statement about any physical system, and it is derived where it belongs rather than here.

What a physicist takes from it is a single conditional: if the sampled dynamics is a one-dimensional map with a smooth maximum, then the constants are those two numbers. Everything below is about the antecedent.

Inside the chaos #

Past the accumulation the picture looks like a smear, and it is not.

The band is threaded with windows: intervals of the knob on which a stable cycle exists again, each opening abruptly, each running its own doubling cascade, each ending in a band. The largest is the three-cycle window, and its existence is not a small fact. A theorem of Sharkovskii orders the integers so that the existence of a cycle of one period forces cycles of every period later in the order, and three is first — a continuous map of an interval with a three-cycle has cycles of every length whatever. That is the content of the phrase “period three implies chaos”, and the window in this figure is where it applies.

The self-similarity is exact in the limit and approximate in the picture. Magnifying the window and rescaling reproduces the whole diagram, with a second Feigenbaum constant — $\alpha = 2.5029$ — describing how much the values have to be rescaled as well as the knob. The two constants together are what the renormalisation operator’s fixed point supplies, and they are the two numbers that anything in this class must share.

Where this does not happen #

It is as important to say which systems do not do this, and the answer separates two of the routes to chaos this collection draws.

Period doubling needs an attractor. A cycle can lose stability to one of twice the length only if there is a notion of stability to lose, and that needs contraction of phase-space volume — dissipation. A frictionless mechanical system has none: Liouville’s theorem says its phase-space volume is preserved exactly, so it has no attractors, and no cycle is stable in the sense used above. Its route into chaos is the destruction of invariant curves as a coupling grows, which has a quite different structure, its own constants, and no cascade of doublings at all.

So the driven damped pendulum period-doubles and the undamped one does not, and the difference is the damping rather than the driving. It is worth being suspicious of any general claim about “the route to chaos” that does not say which of the two it means; the systems in this collection divide cleanly between them, and the number 4.669 belongs entirely to one side.

Both routes share the mechanism underneath, and it is the one drawn in the fold that has to be there. Stretching separates neighbouring states — which is the error doubling on a schedule — and folding brings them back into a bounded region. The map here does both in one line: $rx(1-x)$ stretches near the ends and folds at the middle, and the maximum that makes it fold is the same maximum whose curvature the constant depends on.

What the windows are, and why they matter here #

The band past the accumulation is threaded with windows — intervals of the knob on which a stable cycle exists again, each running a doubling cascade of its own. Their order is not arbitrary: Sharkovskii’s theorem arranges the periods so that the existence of one forces every period after it, with three at the front, and matpic’s account of the diagram works that out.

The experimental consequence is what matters here, and it is a warning rather than a result. A system that has been driven past its accumulation point does not stay chaotic as the drive rises. It passes through windows in which it is perfectly periodic again, at period three or five or six, and a laboratory sweeping a control parameter in steps can land in one and conclude the chaos has gone. The windows are narrow — the largest is a fraction of a per cent of the drive range in the convection experiments — so whether they are seen at all depends on the step size, and two groups sweeping the same apparatus at different resolutions will disagree about where chaos begins.

Where the model stops #

A one-dimensional map is a shadow of a dynamical system, not one. No physical system is a one-dimensional map; what makes the correspondence work is a strongly contracting attractor sampled once per cycle, so that the state after one drive period depends on the state before it through effectively one number. Where the contraction is weak the map is genuinely two-dimensional, the cascade is modified, and the constants shift.

The universality class is narrow and is stated by a derivative. Everything here requires a maximum with a non-zero second derivative. A map with a quartic maximum has its own constants — different numbers, same argument — and a map with a corner rather than a smooth maximum has neither. So “any nonlinear map” is much too broad a claim: the class is defined by the local shape at one point.

And the arithmetic runs out before the sequence does. The gaps shrink by a factor of 4.67 each time, so the tenth doubling occurs in an interval of width around $10^{-7}$ of the original and the twentieth at $10^{-14}$, which is where double precision stops distinguishing settings. The figures compute eight and stop, and the last ratio they report is 4.66919 rather than 4.66920 for that reason and not because the sequence has stopped converging.

What the pictures cannot show #

The cascade diagram draws where an orbit goes and not how fast it gets there. Near a bifurcation the approach to the cycle slows down without limit — critical slowing, the same phenomenon as at a thermodynamic critical point — so the columns near a branching would need far longer runs to be correct, and are drawn with a fixed transient. The visual consequence is a slight blurring at each fork, which is an artefact of the computation rather than a feature of the map.

Nor does the picture distinguish an orbit that visits a band densely from one that visits a few thousand points very close together. Past the accumulation there are settings with genuinely chaotic orbits and settings with stable cycles of very long period, and at the resolution drawn they look identical. The set of knob values giving chaos has positive measure and is riddled with windows in every interval, and a figure with 320 columns can only report what those 320 samples happened to be.

Where the ladder goes next #

The chaos ladder began with an error that doubles on a schedule, went through the last invariant curve to be destroyed and the fold that has to be there. This rung asks how a continuous dissipative system comes to be described by a map at all, which is the condition every universal statement about maps has to clear before it says anything about an experiment. The rungs after it: intermittency, the second route, where a system is nearly periodic for long stretches and bursts irregularly; the strange attractor’s dimension, which is not an integer and is measurable; and control of chaos, where the unstable cycles embedded in an attractor are stabilised by nudges far smaller than the motion.

The habit worth carrying away is to look for the reduction before believing the universality. A theorem about a class of models says nothing about an apparatus until somebody shows the apparatus is in the class, and here the showing is a statement about damping: contract the phase space hard enough and a fluid with infinitely many degrees of freedom becomes one number per drive cycle. Where the contraction is weak the theorem is still true and no longer applies, and the constants measured drift away from it for a reason that is about the equipment rather than about the mathematics.