Doubling without an attractor

A constant that does not care which map found Feigenbaum’s 4.6692 in the logistic map and the sine map alike, and one constant for each shape of top found where that universality ends: maps whose humps are flatter or sharper than a parabola have constants of their own, one for each order of the maximum. Both stayed inside one kind of system — a map that folds an interval onto itself, losing information and, in two dimensions, area as it goes. This essay leaves that kind of system altogether and finds period doubling, with a new constant, where there is nothing to attract an orbit at all.

The map is Michel Hénon’s quadratic map in its area-preserving form,

$$x_{n+1} = 1 - a\,x_n^2 - x_{n-1},$$

which in the plane of pairs $(x_n, x_{n-1})$ is a map with Jacobian determinant exactly 1. Every region keeps its area under it. Systems like this come from mechanics without friction — a pendulum kicked once a period, a particle in a storage ring passing a focusing magnet, a planet’s orbit sampled once a year — and because area is conserved, no orbit can settle onto anything. There is no attracting fixed point, no attracting cycle, no strange attractor. Yet the periodic orbits still double.

Orbits that are encircled rather than attracting #

In a map that keeps area, a periodic orbit has two possible characters. Linearise the map around the orbit, multiply the derivative matrices once round it, and call the product the monodromy matrix. Its determinant is 1, so its two eigenvalues multiply to 1, and what decides the orbit’s character is the trace. If the trace lies between −2 and 2, the eigenvalues are complex numbers on the unit circle, the linearised map is a rotation, and nearby points circle the orbit on closed curves: the orbit is elliptic, the centre of an island. If the trace is outside that range, the eigenvalues are real, one larger than 1 in size and one smaller, and nearby points are flung away along one direction: the orbit is hyperbolic.

The fixed point of the map is elliptic for small $a$. Its trace is $-2ax^*$, and it reaches −2 at exactly $a = 3$. There the eigenvalues are both −1, the island around the fixed point splits, and an elliptic orbit of period 2 appears beside it, its two points on either side of what is now an inverse-hyperbolic fixed point. The period-2 orbit is born with trace +2, its trace falls as $a$ grows, and at $a = 4$ exactly it reaches −2 in turn: the orbit $(\tfrac12, 0)$, whose monodromy matrix the computation evaluates to $-2$ to every digit. Then period 4 is born, and so on.

The figure follows each orbit by Newton’s method on the whole orbit at once: the $N$ unknowns $x_0, \ldots, x_{N-1}$ of a period-$N$ orbit must satisfy $N$ equations, $x_{n+1} + x_{n-1} = 1 - a x_n^2$ with indices taken round the cycle, and Newton’s method solves them all together from a good guess. The doubling point of each orbit is located by bisection on the condition that its trace equals −2, and the new orbit is started there by nudging the doubled old orbit along the eigenvector the doubling releases.

Finding an orbit nothing approaches #

The method has to work without the help the dissipative case gives. In the logistic map’s cascade an attracting cycle can be found by iterating any starting point and waiting; here iteration only ever traces a closed curve around an orbit, never the orbit itself. So the orbit is solved for. For a period-$N$ orbit the $N$ equations form a sparse system — each involves three neighbouring unknowns — and Newton’s method solves the linearised system at each step, converging quadratically once it is close. Close is the hard part. Just after a doubling, the new period-$2N$ orbit lies within a distance proportional to $\sqrt{a - a_k}$ of the old orbit traversed twice, so the computation starts a little past each doubling, perturbs the doubled old orbit by a small multiple of that square root along the eigenvector with eigenvalue −1, and asks Newton’s method to find a genuinely period-$2N$ solution, checking that it has not slid back onto the old orbit of half the period. From there the orbit is followed in small steps of $a$ until its own trace reaches −2.

Near each doubling the Newton matrix becomes nearly singular, since a new direction of solutions is opening up, and the method needs care there; away from the doublings it is fast. Locating the doubling of the period-256 orbit took a few seconds, most of it spent solving 256 linked equations at each of the steps along the orbit’s short life. The twelve figures of accuracy in the doubling parameters are what let the interval ratios be read to six: the last interval is about $3 \times 10^{-7}$ wide, and its ratio to the one before needs both ends known to far better than that.

The ratio is 8.721 #

The doublings come at $a = 3$, 4, 4.1204525, 4.1343639, 4.1359601, 4.1361431, 4.1361641 and on, accumulating at $a = 4.1361668$. The intervals between them shrink, and their ratios — 8.302, 8.659, 8.7156, 8.7203, 8.72103, 8.72109, 8.72110 — settle on

$$\delta = 8.7210972\ldots,$$

nearly twice Feigenbaum’s 4.6692. The constant was found around 1980 independently by Giancarlo Benettin, Carlo Cercignani, Luigi Galgani and Antonio Giorgilli in Italy and by John Greene, Robert MacKay, Franco Vivaldi and Mitchell Feigenbaum in the United States, and it is universal in the same sense as Feigenbaum’s: every smooth area-preserving map of the plane whose orbits double in this way, from a kicked rotor to a model of a particle beam, gives the same 8.721. The class is set by area preservation, as the dissipative class is set by the folding of an interval.

Space shrinks by 4.018 #

Feigenbaum’s cascade has a second constant, $\alpha = 2.5029$, for how much smaller each new cycle is than the last near the top of the hump. The area-preserving cascade has one too. At the moment each orbit doubles, measure the distance between its point on the line of symmetry and the point half a period later; from one doubling to the next the distance shrinks by 4.182, 4.0246, 4.0203, 4.01813, 4.01811, 4.01808 — settling on

$$\alpha = 4.0180767\ldots,$$

with a reversal of orientation at each step that the figure’s magnitudes do not show. There is a second spatial constant in the other direction, about 16.36, and the two together must be consistent with keeping area: a dissipative map can squeeze in one direction as hard as it likes, but here each squeeze in one direction is paid for by a stretch in another. The renormalisation picture that explains Feigenbaum’s constants — zoom in on one cycle of the map’s twice-composed version and find a copy of the original — works here too, with a different fixed point of the zooming and, because the zooming now acts on maps of the plane, two scaling factors rather than one.

Closed curves, not an attractor #

What the doubling looks like is different from the dissipative case, and the phase portrait shows why. In the logistic map, after a doubling every nearby orbit is drawn onto the new attracting cycle and the old one is abandoned. Here nothing is drawn anywhere. At $a = 3.2$, just after the first doubling, points started near the now-unstable fixed point move on closed curves that wind around both points of the period-2 orbit — a figure-of-eight family of curves, each one an orbit’s permanent home. Points started a little further out lie on curves that enclose everything. The picture is a set of nested invariant curves and island chains, the structure that the last circle to break followed in the standard map, and doubling is a change in which curves surround which orbits.

Because no orbit is ever attracted, the doubling cascade cannot be watched by iterating one starting point, as it is in the logistic map’s bifurcation diagram. It has to be found by solving for the periodic orbits directly — which is why every figure in this essay comes from Newton’s method rather than from iteration.

Leaking area moves the constant #

The two universal constants belong to two classes, and the general Hénon map, $x_{n+1} = 1 - ax_n^2 + bx_{n-1}$, connects them. Each step multiplies area by $|b|$, so $b = -1$ keeps area and every other value in $(-1, 1)$ loses it. The same code follows the cascade at five values. At $b = -0.999$ the ratios start 8.302, 8.659, 8.716 and then drift down; at $b = -0.99$ they reach 8.70 and fall to 7.99; at $b = -0.9$ they go 8.21, 8.32, 7.58, 6.07, 4.90, 4.67 — from one constant to the other in six doublings; at $b = -0.3$ they are near 4.67 from the start.

The reason is that a doubling cascade explores smaller and smaller scales of the map, and at small scales even a tiny loss of area per step accumulates over the long periods involved: an orbit of period $N$ loses area by $|b|^N$ once round, and when $|b|^N$ is no longer close to 1 the orbit is genuinely attracting and the dissipative class takes over. A map that leaks one part in a thousand per step follows the area-preserving constant for orbits of period up to a few hundred and then turns. The two constants are the two ends of one crossover, and which one an experiment sees depends on how long its orbits are compared with its friction.

Every orbit lives the same life #

The self-similarity behind the constant can be seen in a single number per orbit. Every orbit is born with monodromy trace +2, the eigenvalue 1 of its birth, travels down through the elliptic range, and doubles at −2. Rescale the parameter so that each orbit’s life runs from 0 to 1 and the traces of periods 2 to 32 trace nearly the same path, almost a straight line from +2 to −2. The orbit of period 32 lives through exactly the same sequence of shapes as the orbit of period 4, on a parameter interval $8.72^3$ times shorter and in a region of the plane about $4.02^3$ times smaller. That is the content of renormalisation: one orbit’s life, repeated at every scale, with two numbers fixing the scale.

The same idea one level down #

The structure of this cascade — an elliptic orbit giving way to an elliptic orbit of twice the period — is one instance of a more general phenomenon in area-preserving maps. An elliptic orbit’s trace measures the rotation of nearby points around it, and when that rotation passes through any rational fraction of a turn $p/q$, not only through one half, a chain of $q$ islands is born around it: the Poincaré–Birkhoff theorem, which a twist that cannot avoid two points proved for the annulus, guarantees that a pair of orbits of period $q$ appears whenever a twist passes a rational rotation. Period doubling is the case $p/q = 1/2$, where the island chain has just two islands and the new orbit is born from the old one itself. Island chains for other rotations sit around every elliptic orbit at every level, which is why the phase portrait is islands within islands, and the doubling cascade is the single thread through that hierarchy that the renormalisation can follow exactly.

The cascade also ends the way the dissipative one does, and in a way it does not. Past the accumulation point $a = 4.1361668$ the orbits of every period $2^k$ are hyperbolic, and the islands around them have shrunk to nothing; but area preservation means the region they leave behind cannot be an attractor, and orbits that wander there fill a chaotic sea of positive area rather than a fractal set of zero area. Hénon’s dissipative map has stable windows inside its chaos because orbits can be attracted into them; the area-preserving map has islands inside its sea instead, which can be entered only by starting there.

Where the map came from #

Michel Hénon introduced the quadratic area-preserving map in 1969, a decade before the dissipative map that carries his name, as a model for the orbits of stars in a galaxy: sampling a star’s motion once per passage through a plane turns a smooth Hamiltonian flow into an area-preserving map of that plane, and the simplest such map with a non-trivial twist is quadratic. His pictures of nested curves, island chains and chaotic seas were among the first images of what is now called the mixed phase space of Hamiltonian chaos, and the period-doubling sequence was visible in them as a cascade of islands within islands, long before anyone measured how fast it converged.

The universal constant came eleven years later, in the wake of Feigenbaum’s discovery for dissipative maps, when several groups asked whether the conservative case had its own. That it does — a different number for a different class — is the conservative counterpart of what the order of the maximum did for one-dimensional maps: universality is real, but it is universality within a class, and what defines the class is which features of the map the zooming of renormalisation cannot remove. Folding, the order of a maximum, and the conservation of area are all such features. The same cascade appears in a flow that doubles like a parabola, where friction puts it in Feigenbaum’s class; take the friction away and the flow would join this one.

What the computations show and assume #

The doubling parameters are computed, not quoted, by Newton’s method on orbits of up to 256 points and bisection on the trace condition, to about twelve significant figures; the last interval ratio, 8.721096, agrees with the published 8.7210972 to six figures, and the last spatial ratio, 4.018078, with 4.0180767 to five. The ratios are convergent sequences, and their limits are the universal constants only in the sense the renormalisation theory establishes, which is a computer-assisted proof for the area-preserving case by Jean-Pierre Eckmann, Hans Koch and Peter Wittwer in 1984; the figures show convergence and do not prove a limit. The phase portrait iterates 26 starting points for 600 steps each and shows that they stay bounded for that long, which is evidence of closed curves rather than proof of them.

Still open: the islands that survive #

Every orbit in the cascade is surrounded, while elliptic, by closed invariant curves and island chains, and the theory of Kolmogorov, Arnold and Moser guarantees that most of those curves survive small perturbations. How much of the plane near the accumulation point is covered by islands, and how much by chaotic sea, is not known: the islands become infinitely nested and infinitely small, and whether their total area has a simple scaling law tied to the constants 8.721 and 4.018 is a question that numerical experiments suggest and no proof settles. The same question, for the islands around the elliptic orbits of a general area-preserving map, is one of the central problems of Hamiltonian dynamics: whether the chaotic sea of the standard map, at any value of its parameter, has positive area is still open.

A second universality #

An area-preserving map has no attractors, yet its elliptic periodic orbits double in a cascade — at a = 3, 4, 4.1205, 4.1344, … in Hénon’s map, accumulating at 4.13617 — and the intervals shrink by 8.721 per doubling and the orbits by 4.018, not by Feigenbaum’s 4.669 and 2.503. The doubling is a change in which closed curves encircle which orbit, so it must be found by solving for orbits rather than by iterating; each orbit is born at trace +2 and doubles at −2, living the same rescaled life at every level. Let the map leak area and the cascade crosses from one constant to the other, later the less it leaks: two universality classes, one for friction and one for its absence.