Stable cycles a hair from Hénon's chaos

A region the orbit cannot leave ended on a question about one map. Michel Hénon’s map sends a point $(x, y)$ of the plane to $(1 - ax^2 + y,\ 0.3x)$, and at $a = 1.4$ every computer that has ever iterated it draws the same thing: a curved band folded into itself, the most reproduced strange attractor after Lorenz’s. Whether that picture is what the map really does is not known. The obstruction is specific. As $a$ varies, the chaotic parameters are interleaved with periodic windows — parameter intervals on which the map’s attractor is a single stable cycle, and every typical orbit eventually settles on it. If a window too thin for any scan contained $1.4$, the attractor there would be a cycle of enormous period, and the chaos every computer has drawn would be a transient that outlasts every computation.

That essay explained why the question is hard. This one does the opposite of explaining: it goes and looks. It hunts the windows near $1.4$, finds the nearest ones a scan can see, measures how wide they are, and proves — by a computation that bounds its own rounding errors — that two of them really are windows. None of this settles Hénon’s question, and the last sections say exactly why not. But the hunt changes the question from an abstract worry about thin windows into a list of measured ones, with their sizes and their distances from $1.4$ written down.

What a window looks like from inside #

At $a = 1.39945$, a change in the fourth decimal place from the classical value, the Hénon map has a stable orbit of period thirteen. Iterate from almost anywhere near the attractor and, after a transient of a few thousand steps that looks indistinguishable from chaos, the orbit lands on thirteen points and visits them in turn for ever. At $a = 1.4001616$ the same happens with nineteen points. The hero figure draws both cycles over the grey attractor at $1.4$, and the striking thing is how unremarkable they look. Their points lie on the attractor’s own folded band, spread along it as a chaotic orbit’s points would be, and a reader shown only the first few hundred iterates would have no means of telling these parameters from the classical one.

That is the whole difficulty in one picture. A stable cycle near a chaotic parameter inherits the chaotic attractor’s shape, because the cycle is built from the same stretching and folding that stretch, fold, and what is left described; it is simply one particular orbit through the fold, which the dynamics at that parameter happens to make attracting. The transient before the orbit finds it is chaotic in every visible respect. So the difference between “chaotic” and “periodic with a long transient” is not a matter of what the attractor looks like. It is a matter of the multipliers of one orbit, and they can be computed.

Finding a window #

A scan finds windows by brute force. For each value of $a$ on a fine grid, iterate the map from $(0.1, 0.1)$ for several thousand steps, then keep iterating and watch for the orbit to return to its starting point to ten decimal places. A chaotic orbit never does; a stable cycle of period $p$ does after exactly $p$ steps.

A return is evidence, not proof, so each one is refined. Newton’s method, applied to the equation $F^p(z) = z$ that says the point $z$ returns after $p$ steps, converges from the returned point to the periodic orbit to full double precision in a handful of iterations. The derivative of $F^p$ along that orbit, a two-by-two matrix, has two eigenvalues, the orbit’s multipliers. The orbit attracts exactly when both lie inside the unit circle. For the Hénon map their product is fixed: each step multiplies areas by $-0.3$, so the product of the multipliers of any period-$p$ orbit is $(-0.3)^p$, which for $p = 13$ is about $-1.6 \times 10^{-7}$. One multiplier is therefore nearly nought, and the other is nearly the trace of the matrix. Stability comes down to one number lying between $-1$ and $1$.

Running this over the 400,001 parameters from $1.39$ to $1.41$, spacing $5 \times 10^{-8}$, finds seven primary windows, with periods 9, 10, 12, 13 (twice), 19, and 9 again. Primary means the window begins with the birth of its orbit rather than the doubling of a shorter one: every window is followed by its own cascade of period doublings, and those later orbits belong to the same window rather than to new ones. The nearest window to $1.4$ has period 19 and starts at $a = 1.40016156$. The nearest on the other side has period 13 and ends at $1.39946054$, about $5.4 \times 10^{-4}$ below. Between them, an interval of width $7 \times 10^{-4}$ containing the classical parameter, the scan finds nothing at all.

The life of one window #

Each window has a precise beginning and end, found by following its orbit as $a$ changes. Newton’s method is restarted at each new parameter from the previous orbit, and the trace of the derivative is recorded until a multiplier reaches the unit circle.

The period-13 window runs from $a = 1.3994496192$ to $1.3994605440$, a width of $1.09 \times 10^{-5}$. At its left end the orbit is born together with an unstable partner in a saddle-node bifurcation: one multiplier is exactly $+1$, and just past the birth the trace falls steeply below one, with the square-root speed that how a lock comes apart found at every saddle-node. In the middle the trace passes through nought, at $a = 1.39945235$; there one point of the orbit sits where the fold is tightest, the derivative is as small as it can be, and the orbit attracts its neighbours fastest. At the right end the trace reaches $-1$, the orbit loses its stability by doubling, and a period-26 orbit takes over to begin the cascade.

The shape is the same as the one the window that opens with a stutter traced for the logistic map’s period-three window, compressed by a factor of about a thousand. That essay found the long, nearly periodic bursts just before a window opens — intermittency, the trace of an orbit about to be born. Here the same structure occupies ten millionths of a unit of $a$, and a scan whose spacing were coarser than that would step clean over it.

How wide a window can be #

The windows near $1.4$ are all thin, and the thin ones are the long ones. A scan of 1,200,001 parameters from $1.3$ to $1.42$ finds forty-two primary windows, with periods from 8 to 110, and their widths fall as their periods grow.

The median window of period ten or less is $5 \times 10^{-5}$ wide; the median window of period nineteen or more is $8 \times 10^{-7}$, nearly seventy times thinner. Period alone does not fix the width — the seven windows of period 13 differ in width by a factor of 149 — because what matters is how much the orbit is stretched as it goes round. For one-dimensional maps there is a rule of thumb, proved in some cases: a window’s width is roughly the inverse square of the product of the derivatives along its orbit, excluding the point at the fold. An orbit that passes through strongly stretching parts of the attractor has a large product and a tiny window. Since a typical orbit of the Hénon map at $1.4$ stretches by a factor of about $e^{0.42}$ per step, the product for a long orbit grows exponentially with its period, and the window shrinks exponentially. The scatter in the figure is the difference between orbits that pass close to the fold several times, and so stretch less, and those that do not.

The consequence is the reason the question at $1.4$ is hard. The rule of thumb predicts widths falling by a factor of more than two for each unit of period. The measured windows fall more slowly than that, because long orbits that linger near the fold stretch less than a typical orbit, but not one window found here with a period above 26 is wider than a millionth of a unit of $a$, and the single window of period 110 is $1.7 \times 10^{-8}$ wide. A scan of uniform spacing cannot find windows thinner than its spacing, and the windows that could hide near $1.4$ are thinner than any spacing a computer can afford.

Looking closer #

The obvious response is to look harder near $1.4$. The next figure records four scans centred on the classical parameter, each of 20,001 parameters and each ten times narrower and ten times finer than the last.

The widest scan, out to $\pm 0.01$ at a spacing of a millionth, finds seven windows but misses the period-19 window, which at $1.5 \times 10^{-7}$ is thinner than that spacing; it reports the period-13 window as the nearest. The next, $\pm 0.001$ at a spacing of $10^{-7}$, finds the period-19 window as well. Inside $\pm 10^{-4}$, at a spacing of $10^{-8}$, and inside $\pm 10^{-5}$, at $10^{-9}$, the scans find nothing. A further scan not drawn here, of 400,001 parameters within $2 \times 10^{-5}$ of $1.4$ at a spacing of $10^{-10}$, iterating each 20,000 times and watching for periods up to 3,000, found nothing either.

What this establishes is narrow and should be stated precisely. Near $1.4$ there is no window of period up to a few thousand whose width exceeds the spacing of the scan that looked there, provided the scan’s transient was long enough for the orbit to settle. That is a statement about what is absent at one resolution. It is perfectly consistent with a window of period 200 and width $10^{-30}$ sitting exactly on $1.4$, and the exponential shrinking of the previous section says that, if windows are dense, such windows exist arbitrarily close to it.

Proving a window is a window #

A floating-point computation that finds a stable orbit is persuasive but not a proof. Rounding error enters every step, and the orbit a computer draws showed how badly a chaotic orbit’s arithmetic can mislead. A window, however, can be proved with a finite computation, because a stable orbit is a robust object: it exists in a neighbourhood and attracts everything in it. The tool is interval arithmetic, in which every number is replaced by an interval guaranteed to contain it, and every operation is rounded outward so that the guarantee survives.

The test used here is Rudolf Krawczyk’s, from 1969. Put a small box around each of the thirteen points Newton’s method found, and treat the whole orbit as one unknown: thirteen points, each the image of the one before. Krawczyk’s operator takes the boxes, the equations and an approximate inverse of their derivative, and produces a new set of boxes. A theorem guarantees that if the new boxes land strictly inside the old ones, the equations have exactly one solution in them: there is exactly one orbit of period 13 passing through the boxes.

For the period-13 orbit at $a = 1.39945$, boxes of half-width $10^{-10}$ around each point are mapped into boxes of half-width $3.7 \times 10^{-12}$, nearly thirty times smaller, so the orbit exists. Over the same boxes, interval arithmetic bounds the trace of the derivative of the thirteenth iterate between $0.62634$ and $0.62635$ — comfortably inside the band $\pm(1 + \det)$ that stability requires, with the determinant $(-0.3)^{13}$ known exactly. So the orbit attracts, and $1.39945$ lies in a window. For the period-19 orbit at $a = 1.4001616$ the boxes are mapped into half-width $9.4 \times 10^{-12}$ and the trace is bounded between $-0.1027$ and $-0.1026$: that orbit too exists and attracts. The parameter itself is carried as an interval around its decimal value, so the statements are about the numbers $1.39945$ and $1.4001616$ as written, not about the nearest numbers a computer can store.

One detail matters for the period-19 orbit. Pushing a box through all nineteen steps of the map at once, the first approach tried here, fails: each step stretches the box by up to a factor of nearly three, interval arithmetic cannot exploit the cancellations that keep the true orbit tame, and after nineteen steps the box is too wide to be mapped inside itself. Treating the nineteen points as separate unknowns, so that each box passes through a single step, is what made the certificate possible. It is the same trick, multiple shooting, that serious computer-assisted proofs about chaotic maps use for long orbits, and the reason they can certify periods in the hundreds.

What has been proved about Hénon’s map #

Hénon introduced the map in 1976 as a two-dimensional caricature of the Lorenz equations’ return map, and chose $a = 1.4$ and $b = 0.3$ because the picture looked right. Since then the mathematics has advanced around the classical parameters without reaching them. Michael Benedicks and Lennart Carleson proved in 1991 that for small $b$ there is a set of $a$ of positive measure at which the map has a strange attractor; their parameters are near $a = 2$ and $b$ far below $0.3$. Sheldon Newhouse had shown in the 1970s that near any parameter where stable and unstable directions are tangent, there are parameters with infinitely many coexisting stable cycles. And computer-assisted proofs have established, at the classical parameters themselves, that the map has positive topological entropy: its periodic orbits, counted by period, grow exponentially in number, so there is chaos somewhere in it.

In 2015 Zbigniew Galias and Warwick Tucker asked the question in the title of a paper — Is the Hénon attractor chaotic? — and attacked it in the way this essay does, with far more computing: a systematic search for periodic windows near the classical parameters, with the stable orbits in them certified by interval arithmetic. They found many windows close to the classical parameters and none containing them, and no search of that kind could have proved that none does. The present computation is a small version of theirs, and its two certified windows sit inside the picture they drew.

Still open: whether 1.4 is chaotic #

It is unknown whether the Hénon map at $a = 1.4$, $b = 0.3$ has a strange attractor or a stable periodic orbit. The computation here sharpens what an answer would require without supplying one. Either outcome is consistent with every measurement: a window could contain $1.4$ if its period were large enough to make it thinner than every scan that has looked, and the attractor could be strange if $1.4$ belongs to the positive-measure set of chaotic parameters that, by analogy with Benedicks and Carleson’s theorem, presumably exists near it too.

A proof of chaos would need a robust property that no periodic window can share — something like the expansion along every orbit of a set that the attractor cannot escape — and would need it at exactly $1.4$, since nearby parameters can be periodic. A proof of periodicity would need luck: a window found by search that happens to contain the classical value, and then a certificate like the ones above. Neither has been found. Whether the windows are even dense in the Hénon family, as they are for the logistic map by the theorem of Graczyk, Świątek and Lyubich, is itself open in two dimensions.

A list of thin windows #

A region the orbit cannot leave proved things about a trapping region and about the Lorenz flow. This essay has proved two things about two parameters: at $a = 1.39945$ and at $a = 1.4001616$ the Hénon map has an attracting periodic orbit, of period 13 and 19, and the computer’s picture of chaos at those values is a transient. Nothing about $1.4$ itself has been proved. But the measurement has moved the question. The nearest window found lies $1.6 \times 10^{-4}$ from the classical value and is $1.5 \times 10^{-7}$ wide; nothing nearer was found at any resolution down to $10^{-10}$; and the windows that remain possible are those of long period, which the width figure says are exponentially thin. If $1.4$ is periodic, it is periodic in a window of a size no simulation will ever see, and the sensitivity that makes the map chaotic is the same stretching that makes such a window thin.