A point that pulls, and a point that pushes

A fixed point is a value the rule returns unchanged: $f(x) = x$. On the graph it is a crossing of the curve with the diagonal, and finding one is a matter of solving an equation.

Whether an orbit ever gets to one is a completely different question, and the answer is not in the equation. It is in the slope.

The crossing has not moved between the panels — it is at $1 - 1/r$ in both, and it exists for every $r$ above one. What has changed is how steeply the curve passes through it, and that is what decides everything.

Why the slope is the whole answer #

Zoom in on a crossing until the curve looks straight. Near a fixed point $x^\*$ the map is approximately

$$f(x) \approx x^\* + m\,(x - x^\*),$$

where $m = f'(x^\*)$ is the slope there. Write $d$ for the distance from the fixed point. One step multiplies $d$ by $m$; two steps multiply it by $m^2$; $n$ steps by $m^n$.

So the whole question is what powers of $m$ do. If $|m| < 1$ they shrink to nothing and the orbit converges. If $|m| > 1$ they grow and the orbit leaves. The threshold is exactly one and there is nothing arbitrary about it: one is the slope of the diagonal, and the diagonal is what the map is being compared against.

That is the entire theory of stability for a one-dimensional map, and it takes four lines because the linearisation is honest — near enough to the fixed point, a smooth map really is its own tangent line.

Reading it off the picture #

The algebra is short and the picture is shorter.

The sign of the slope is visible too, and it says something the size does not. A positive slope gives a staircase that approaches from one side, monotonically. A negative slope gives one that overshoots each time and comes back — the spiral above. Both converge when the size is under one; they simply converge differently, and an orbit that alternates about its limit is a different experience from one that creeps up on it.

Watching those two side by side is the argument. Nothing about the crossing changed. The map’s steepness there crossed a threshold, and a point that was pulling started pushing.

The logistic map’s own arithmetic #

For the map $f(x) = rx(1-x)$ the whole thing is computable in closed form, which is unusual and worth using.

The fixed points are $0$ and $x^\* = 1 - 1/r$. Differentiating gives $f'(x) = r(1 - 2x)$, so the slope at the origin is $r$ and the slope at the other crossing is

$$f'(x^\*) = r\left(1 - 2\left(1 - \tfrac1r\right)\right) = 2 - r.$$

Both facts fall out immediately. The origin is attracting for $r < 1$ — a population that dies out — and repelling above it. The other fixed point is attracting when $|2 - r| < 1$, which is $1 < r < 3$, and repelling above three. Every figure in this essay checks both against the drawn orbit.

That is why $r = 3$ appears where it does. It is not a value anybody chose; it is where $2 - r$ reaches $-1$, and it is the last parameter at which this map has anything to settle on.

What happens exactly at the threshold #

The interesting case is the one the inequality leaves out: $|m| = 1$ exactly.

Linearisation says nothing there. One step multiplies the distance by something of size one, so the tangent line predicts an orbit that neither approaches nor leaves — and the tangent line is not the map. What actually happens is decided by the next term in the expansion, the curvature, and it can go either way.

This matters practically as much as theoretically. Near a stability threshold, convergence becomes arbitrarily slow, so a system observed for a fixed length of time looks unsettled long before it becomes unstable. “It has not settled yet” and “it will never settle” are different claims, and near the threshold no finite observation separates them.

The name for a parameter where a fixed point changes stability is a bifurcation, and this one — where the slope passes through $-1$ and a two-cycle appears — is a period-doubling bifurcation. It is the first of infinitely many, which is the next rung of this ladder.

The cycle has a slope too #

A two-cycle is a fixed point of the map applied twice, so the same test applies to it — and applying it says where the next threshold will be.

If $a$ and $b$ are the two points of the cycle, the slope of $f \circ f$ at $a$ is, by the chain rule, $f'(b)\,f'(a)$: the product of the slopes at the two points. The cycle attracts exactly when that product is smaller than one in size.

The product starts near zero when the cycle first appears, grows as $r$ rises, and eventually reaches $-1$ — at which point the two-cycle stops attracting and a four-cycle appears. Then the four-cycle’s slope, a product of four numbers, does the same thing.

So the mechanism repeats itself, and it repeats on an object that is twice as complicated each time. That is the reason the doublings keep coming, and it is also the reason they come faster and faster: a product of many slopes passes through $-1$ over a shorter range of $r$ than a product of few.

The same criterion, everywhere #

The slope test is not about the logistic map or about this field. It is the criterion for any iterative process, and this collection has already relied on it twice without naming it.

Newton’s method is the map $x \mapsto x - f(x)/f'(x)$. Its fixed points are the roots of $f$, and its derivative at a root is zero — which is a slope not merely under one but as favourable as it can be. That is precisely why Newton’s method converges so fast: the distance is not multiplied by a constant, it is squared, so the number of correct digits doubles each step.

Euclid’s algorithm is the same story in whole numbers. Each step strictly shrinks the smaller side, which is a contraction, and a contraction on a set with no infinite descending chain must terminate. The geometric version — the leftover strip always smaller than the one before — is the slope test with the analysis removed.

The general statement is the contraction mapping theorem: a map that shrinks every distance by a fixed factor has exactly one fixed point, and every orbit converges to it. It is one of the few theorems that gives existence, uniqueness and a method of finding it in the same sentence.

A fixed point worth computing #

The test is not only diagnostic. Turning an equation into a map whose fixed point is its solution, and then iterating, is a method — and the collection already has its best example.

The golden ratio satisfies $\varphi = 1 + 1/\varphi$. Read the right-hand side as a map, $g(x) = 1 + 1/x$, and $\varphi$ is its fixed point. The slope there is $-1/\varphi^2 = -0.382$, comfortably inside the interval, so iterating from almost any positive start converges — and the successive values are $1, 2, 3/2, 5/3, 8/5$, which are the Fibonacci fractions.

That is the same sequence the continued fraction produces, arrived at as an orbit rather than as a peeling, and the slope of $-0.382$ is a quantitative statement about it: each step cuts the error to about two fifths, so the convergents gain roughly one bit of accuracy per step. The slowness of the golden ratio’s approximation, which the number field measured with Hurwitz’s theorem, is this slope seen from the other side.

The general recipe is worth naming because it is how a great deal of numerical work is actually done: to solve $F(x) = 0$, invent a map whose fixed point is the root, check its slope, and iterate. Newton’s method is the choice of map that makes the slope zero.

What it does not give #

The contrast with Brouwer’s theorem is the useful part of this essay, and it is a contrast between two kinds of guarantee.

Brouwer says a continuous map of a disc to itself has a fixed point. It needs no slope, no smoothness, no contraction — only continuity and the right shape of domain. And it hands over nothing: the proof is topological, it produces no point, and it says nothing about whether iterating the map would find one.

The slope test says the opposite thing in both directions. It needs smoothness and a fixed point already in hand, and in exchange it says whether iteration will get there and how fast. One theorem is about existence and gives no method; the other is about method and assumes existence.

Most of the useful work in this field lives in the gap. A system can have a fixed point that nothing ever reaches — the right panel of the first figure is exactly that — and calling such a point an equilibrium is true and misleading, because nothing observed will ever be found at it.

Several fixed points at once #

A map with more than one crossing has more than one fixed point, and each gets its own verdict from its own slope.

When every fixed point repels, the orbit is forced into whatever else the map offers: a cycle, or nothing periodic at all. That is not a rare configuration. It is what “chaotic” looks like from the point of view of this essay’s criterion — every fixed point and every cycle repelling, so the orbit is pushed off each of them in turn and never lands.

Which is worth stating plainly, because it inverts the usual intuition. Chaos is not the absence of structure. It is a great deal of structure, all of it unstable.

What a physicist calls it, and what changes #

The same criterion is called several things, and the names are worth knowing because they are how the idea reaches other subjects.

An engineer calls a fixed point with every slope under one asymptotically stable, and the analysis of whether a control system settles is exactly this test applied to the loop’s Jacobian. An economist calls it a stable equilibrium and asks whether a market perturbed slightly returns. A biologist modelling a population asks whether a species knocked off its carrying capacity comes back or oscillates — which is the logistic map’s own original setting, and the reason $r$ in these figures is a growth rate.

What none of those renamings changes is the mathematics: a number is compared against one. What they change is what the number means, and therefore what the threshold costs. Passing $r = 3$ in a fisheries model is a population that starts alternating between good years and bad ones for reasons internal to the arithmetic rather than external to the fishery, which is a genuinely counter-intuitive thing to have to explain to somebody looking for a cause.

That is the standing use of this field: it supplies causes that are not events. A system can change its behaviour completely with nothing happening to it, because a parameter drifted past a threshold that nothing marks.

The basin, and its edges #

Attracting is a local property, and the natural next question is how far it reaches. The set of starting points whose orbits converge to a given fixed point is its basin of attraction, and for a one-dimensional map with a single hump the basins are usually intervals with tidy ends.

Usually. The basin’s boundary consists of points that go to neither — typically the repelling fixed points and their preimages — and those form a set that can be very far from tidy.

In one dimension the mess is limited, because there is only one direction to be pushed in. In the plane it is not limited at all: the basin boundary of a perfectly reasonable map can be a curve of infinite length, and every point on it can have every basin arbitrarily close by. That is what Newton’s method on a complex cubic does, and it is one of the places where a picture is the only reasonable way to state the result.

More than one dimension #

In the plane the same criterion holds with the slope replaced by a matrix, and the replacement brings in a piece of machinery this collection has already built.

Near a fixed point, a smooth map of the plane is approximately its Jacobian — the matrix of partial derivatives — and iterating the map is approximately iterating that matrix. What a matrix does under repetition is decided by the directions it leaves alone and the factors it stretches them by: its eigenvectors and eigenvalues.

So the test becomes: the fixed point attracts when every eigenvalue is smaller than one in size. If one exceeds one and another does not, the point attracts along one direction and repels along another — a saddle, which has no one-dimensional analogue and is the configuration most of the interesting behaviour in the plane is built from.

That is also where the tidy one-dimensional picture ends. A saddle’s stable and unstable directions are curves rather than lines once the map is nonlinear, and when those curves cross each other transversally the result is a tangle that Poincaré found in 1890 and declined to draw, writing that he would not even attempt it. The Lorenz system is what one looks like when a computer draws it instead.

Where the ladder goes next #

This is the third rung of the fixed-point ladder. Brouwer gave existence with no method; the cobweb gave a way to see an orbit; this rung says which fixed points an orbit will actually find.

The next question is what happens after the slope passes $-1$. The fixed point stops attracting and a two-cycle appears — and the two-cycle has a slope of its own, which eventually passes $-1$ too. Following that all the way is the road paved with doublings, and it ends somewhere finite, which is the surprising part.