The dark lines are one point's orbit

The period-doubling cascade ends at a parameter near $3.5699$, and to the right of it the bifurcation diagram stops being a tree of branches. It becomes grey: bands filled with points, sometimes two, sometimes four, sometimes one, interrupted by white windows where a cycle takes over. Anyone who has drawn it knows the other feature, which is less often explained. The grey is not uniform. It is crossed by darker curves — sharp lines that arc through the bands, meet, cross, and run into the edges.

Those curves look like artefacts of plotting, and they are not. They are the most structured thing in the chaotic region, and they all come from one point.

The map is $f(x) = rx(1-x)$, a hump over the interval from nought to one, highest at $x = 1/2$. That top of the hump is the critical point — the one place the map’s slope is nought, and so the one place where two different inputs, one each side, are sent to the same output. The dark curves are its orbit.

The first two images are the edges #

Start with the simplest claim. Once an orbit has taken a single step, it can never be higher than the top of the hump, which is $f(1/2) = r/4$. Everything the map produces is at most that value. And the lowest value the map produces from inputs in $[0, r/4]$ is the image of the endpoint $r/4$ itself, which is $f(r/4) = f^2(1/2)$.

So the interval from $f^2(1/2)$ to $f(1/2)$ is trapped: an orbit that enters it stays in it, and every chaotic orbit enters it within a couple of steps. Its two ends are the first two images of the critical point, and they are the top and bottom edges of the grey region, at every parameter past the cascade.

The magnified strip shows the other images doing what the first two do at the edges. The third and fourth images mark the inner edges of the gap between the two bands at the left; the higher images wander through the grey and every one of them sits on a visible dark line. The first and second images bound everything; the rest bound or darken what is inside.

Why a single point leaves a line #

The reason a single point’s orbit should be visible in a cloud of millions is a fact about folding.

Near the critical point the map is almost exactly a parabola: $f(1/2 + \epsilon) \approx r/4 - r\epsilon^2$. So a small interval of width $2\epsilon$ around $1/2$ is squashed into an interval of width about $r\epsilon^2$ just below $r/4$. Turn that round. The points landing within $\delta$ of the top value come from an interval of width about $2\sqrt{\delta/r}$ around the critical point — a much wider interval than $\delta$ when $\delta$ is small.

A fold concentrates. If the orbit visits the neighbourhood of $1/2$ with some ordinary density, then the neighbourhood of $f(1/2)$ receives those visits squeezed into a sliver whose width is the square of theirs, and the density there blows up like $1/\sqrt{\delta}$. The next step carries that spike to $f^2(1/2)$, the step after that to $f^3(1/2)$, and so on. Each time the spike is spread by the slope of the map along the way, so later spikes are weaker, but each is still an infinite spike in the limit, sitting at an image of the critical point.

That is what the bifurcation diagram is drawing. A column of the diagram is a histogram turned on its side and shaded by density. Where the histogram spikes the column is dark, and the spikes are at the images of $1/2$. As $r$ changes the images move, and the dark points in neighbouring columns join into curves. The dark lines are the loci of the density’s singularities, and those are the critical orbit.

Why the later lines are fainter #

The histogram’s spikes are not equally tall, and the diagram’s curves are not equally dark. The first two images are sharp edges; the third and fourth are clear lines; by the eighth the curve is visible only because it is drawn. The fold argument predicts the fading and says how fast it goes.

A spike of the form $c/\sqrt{\text{distance}}$ at $f^k(1/2)$ is carried by one more application of the map to $f^{k+1}(1/2)$. Near a point where the map’s slope is $s$, distances are multiplied by $|s|$ and densities are divided by it, so a singularity $c/\sqrt{\delta}$ becomes, at the image, $c/\big(|s|\sqrt{\delta/|s|}\big) = \big(c/\sqrt{|s|}\big)/\sqrt{\delta}$. Each step weakens the spike’s coefficient by the square root of the slope it passes through. After $k$ steps the coefficient has been divided by the square root of the product of the slopes along the critical orbit — the square root of the derivative of $f^k$ at the first image.

In the chaotic region that derivative grows exponentially, at the rate set by the orbit’s Lyapunov exponent, so the spikes fade exponentially along the orbit. A handful of images are visible and the rest dissolve into the grey. At a parameter where the critical orbit returns close to $1/2$ the product of slopes passes through a near-zero factor, the next spikes are strengthened rather than weakened, and a faint line suddenly darkens — which is what the diagram does on the approach to every window, where a curve brightens just before it becomes the cycle.

The same arithmetic explains the band-merging picture. At a Misiurewicz parameter every image from some point on is the same point, so the infinitely many weakened spikes pile onto one location, and their coefficients form a geometric series that converges. The spike there is no taller in kind than the others; there are simply fewer places for the density to be singular, which is why the histogram at a merging parameter looks so much cleaner than the one at $3.8$.

The whole interval, and the two spikes it keeps #

At $r = 4$ the picture is at its simplest, and it connects to a histogram already drawn elsewhere.

At $r = 4$ the top of the hump maps to 1, and 1 maps to 0, which is a fixed point. The critical orbit reaches the end of the interval in two steps and stays there, so there are only two images and two spikes, at the two ends. The density between them is the arcsine distribution, $1/\big(\pi\sqrt{x(1-x)}\big)$, which blows up at exactly those two points like $1/\sqrt{\text{distance}}$ — the square-root pile-up of the fold, in closed form. At this parameter the map is a disguised doubling, and the formula can be written down; at every other chaotic parameter the spikes are still there and the formula is not.

Where the bands merge #

Just past the end of the cascade the chaotic orbit is confined to many narrow bands, and it hops between them in a fixed cyclic order. As $r$ increases the bands widen and join in pairs — sixteen into eight, eight into four, four into two, and finally two into one. The diagram shows each merger as a place where two grey regions touch.

The critical orbit says exactly where. The logistic map has a fixed point $x^* = 1 - 1/r$ in the middle of the interval, unstable once $r > 3$. The two bands at the far right of the cascade sit one on each side of it, separated by a gap whose inner edges are the third and fourth images of the critical point. The bands merge at the parameter where that gap closes — where $f^3(1/2)$ lands exactly on the unstable fixed point, and $f^4(1/2)$, being its image, lands there too.

That is the point in the magnified strip where every curve meets: from the third image on, the critical orbit sits on $x^*$ for ever. A parameter where the critical point’s orbit lands on an unstable cycle after finitely many steps is called a Misiurewicz parameter, and it is where the density’s spikes pile on top of one another.

The same condition, one level down, gives every other merger. Four bands become two when the critical orbit, after five steps, lands on the unstable cycle of period two; eight become four when after nine steps it lands on the unstable cycle of period four. In general $2^n$ bands merge into $2^{n-1}$ where

$$f^{\,2^n+1}(1/2) \;=\; f^{\,2^n+1+2^{n-1}}(1/2),$$

an equation in $r$ alone, solvable by bisection to as many digits as the arithmetic will carry.

A second cascade, running the other way #

The merging parameters crowd to the left, toward the same point the doublings crowded toward from the other side, and the ratios of their gaps are 4.840 and then 4.652. That is Feigenbaum’s constant, $4.6692$, approached from above just as the doublings approach it from below.

The coincidence has the same explanation as the constant itself. The period-doubling picture is self-similar because a map two doublings in looks, after rescaling, like the map one doubling in; the rescaling is an operation on the space of humped maps, and near its fixed point it stretches one direction by $\delta$ per application. The doublings are where a family crosses one sequence of surfaces in that space, and the mergings are where it crosses another sequence on the far side of the fixed point. The same stretching rate governs both, so the reverse cascade in the chaotic region is the doubling cascade seen in a mirror: two bands are one band of the map’s second iterate, four are one band of its fourth, and each merger is a doubling of the bands rather than of the points.

Between them, the two cascades squeeze the accumulation point from both sides. The doublings at $3.5699$ from below; the mergings at $3.57099$, $3.5748$, $3.5926$ from above. Everything strictly between two consecutive mergings has the same number of bands, and the orbit hops round them in the order the corresponding periodic orbit did before the chaos began.

The windows are where the curves pass the top #

The critical orbit’s curves also explain the white windows, and they do it by passing through the one place they have not been mentioned yet: the critical point itself.

Inside a window the attractor is a stable cycle, and the images of the critical point converge to it, so the curves are drawn along the cycle’s points. Where one of them passes exactly through $x = 1/2$ — where $f^k(1/2) = 1/2$ — the cycle contains the critical point, its multiplier is nought, and it is superstable. Every window contains exactly one such parameter for each cycle, which is how windows are located in practice rather than by sampling, and why a point that pulls and a point that pushes singles out the superstable parameters as the diagram’s skeleton.

So the critical orbit does three different things at three kinds of parameter. At a superstable parameter it returns to its start and the orbit is a cycle. At a Misiurewicz parameter it lands on an unstable cycle and the bands merge. At a generic chaotic parameter it wanders, and wherever it wanders the density spikes. Almost everything visible in the chaotic half of the diagram is one of those three.

Why the critical point runs everything #

It is worth asking why one point should have this much authority, and the answer is a theorem about what can attract.

For a map like the logistic one — smooth, with one hump and a negative Schwarzian derivative, which $rx(1-x)$ has — every attracting cycle attracts the critical point. Singer’s theorem of 1978 says a cycle’s immediate basin must contain a critical point or reach the end of the interval, because otherwise the map would be monotone on that basin and a monotone map with a negative Schwarzian derivative cannot hold an attractor there. Since there is only one critical point, there is at most one attracting cycle at each parameter, and to find it one iterates $1/2$ and waits.

That is the reason every bifurcation diagram in this collection starts its orbits near the middle and gets the right answer. It is also why the chaotic region’s structure is legible at all. The single critical orbit is a finite-dimensional thread running through an infinite-dimensional object — the invariant density — and it carries all the information about where that density is singular, where the attractor ends, and when the bands join. Kneading theory makes this precise by encoding the critical orbit as a sequence of left and right symbols and reading the map’s entire topological behaviour, entropy included, off that one sequence.

The same thread in the complex plane #

The logistic family is one real slice of a family that lives more naturally in the complex numbers, and there the role of the critical orbit is not merely convenient but definitional.

A change of variable turns $rx(1-x)$ into $z^2 + c$ with $c = r/2 - r^2/4$, and the critical point $1/2$ becomes $0$. The Mandelbrot set is defined as the set of $c$ for which the orbit of $0$ stays bounded — the critical orbit again, now deciding membership outright. The real parameters of the logistic map from 1 to 4 correspond to the stretch of the Mandelbrot set’s real axis from $1/4$ down to $-2$.

Every feature of the chaotic region has a counterpart there. The period-doubling cascade is a chain of discs along the real axis, each bulb twice the period of the last, shrinking by $\delta$. The windows are the real sections of small copies of the whole Mandelbrot set sitting on the axis — the period-three window cuts through the largest of them, near $c = -1.75$. The superstable parameters are the centres of those copies. And the band-merging parameters are Misiurewicz points: parameters on the boundary where the critical orbit lands on a repelling cycle, which are known to be dense in the boundary of the whole set and are where its filaments branch.

So the real diagram’s dark curves, joints and windows are a one-dimensional section through a two-dimensional structure organised entirely by one orbit. That is why results about the logistic map — Jakobson’s, Lyubich’s, the density of windows — were proved by complex methods: the critical orbit is easier to control where it has room to move.

What the curves cannot show #

The histograms are counts over two million steps of one orbit started at one point, and the diagram is sixty to ninety settled values per column. Both are finite samples of an object — the invariant density — whose existence at a given parameter is itself a theorem that does not always hold. At a parameter inside a tiny window the long-run histogram is three or six spikes and nothing else, and a computed histogram at a parameter lying in a window narrower than the column spacing looks exactly like a chaotic one.

The spikes the figures draw are finite because bins are finite. The density’s singularities are infinite, of the form $1/\sqrt{\text{distance}}$, and no histogram can show an infinite value or distinguish one singularity strength from another. The claim that each image of the critical point carries a genuine singularity is an argument about folds, not something the pictures establish.

And the curves are drawn smoothly across the whole range, including through windows where they are not dark lines of any density but the positions of a cycle. The same curve is a singularity in one column and a periodic point in the next, and the drawing gives no sign where one role ends and the other begins.

Still open: which kind a given parameter is #

For the logistic family the parameters split, as far as anybody can tell, into two kinds. At a regular parameter there is an attracting cycle and almost every orbit is eventually periodic; at a stochastic parameter there is an absolutely continuous invariant density of the kind the histograms approximate, with spikes at the critical orbit. Jakobson proved in 1981 that the stochastic parameters have positive total length; Graczyk, Świątek and Lyubich proved in 1997 that the regular parameters are dense; and Lyubich proved in 2002 that almost every parameter is one kind or the other.

What nobody can do is say which kind a given parameter is. The windows are dense, so every interval of parameters, however small, contains regular ones, and a parameter such as $r = 3.8$ might lie in a window narrower than any computation can resolve. The histogram drawn at 3.8 is overwhelming evidence that the orbit behaves chaotically for two million steps; it is not a proof that it does so for ever, and no general method is known for deciding the question at a specified value.

One point, followed #

The bifurcation diagram looks like a record of every orbit at every parameter, and its chaotic half looks like noise with some unexplained scratches in it. It turns out to be almost entirely determined by following one point. The top of the hump, iterated, bounds the attractor with its first two images, darkens the bands with the rest, closes the gaps between bands when it lands on an unstable cycle, and fixes the windows when it comes back to itself.

The scratches are the structure. Every column is a density whose singularities sit on the critical orbit, and the joints where those curves meet are where the bands merge — at parameters whose gaps shrink by the same constant, on the far side of the cascade, that measured the doublings on the near side.