Two lobes and no cycle

Everything in this field so far has iterated a rule in discrete steps. The Lorenz system is continuous — three differential equations, integrated forward in time — and it is where the word attractor acquired its adjective.

$$\dot x = \sigma(y - x), \qquad \dot y = x(\rho - z) - y, \qquad \dot z = xy - \beta z.$$

Three variables, three parameters, and every term is either linear or a product of two variables. There is nothing in it that a first course would not recognise.

The trajectory winds around one lobe for a while, crosses over, winds around the other, crosses back. How many turns before it switches is not predictable, and the sequence of lobe visits is, as far as anyone can measure, a fair coin.

Every part of that sentence is a claim about a system with no randomness in it. The equations are integrated with a fixed step from a fixed starting point, and running the computation again reproduces the same picture exactly. What is unpredictable is not the arithmetic but the answer to any question about it that is asked far enough ahead.

Where the equations came from #

They are not a model of anything in particular, and knowing that changes how to read them.

Lorenz took a standard model of a fluid heated from below — convection rolls, the mechanism behind everything from a saucepan to the atmosphere — expanded it in a series, and truncated brutally to three terms. The truncation is not physically justified; keeping three modes out of infinitely many is a decision about tractability, not about fluid dynamics.

So the system is a caricature, and the point of it is that a caricature this severe still cannot be solved. If three modes already do this, the full problem is not going to be tamer — which is the argument the system was actually making, and it is an argument about weather forecasting rather than about fluids.

The parameters carry the physics. $\sigma = 10$ is the Prandtl number of the fluid, $\beta = 8/3$ comes from the geometry of the roll, and $\rho$ measures how hard the fluid is being heated. The first two are held fixed by convention; $\rho$ is the knob.

Contracting and stretching at once #

The property that makes this object strange is a pair of facts that sound incompatible.

Volumes shrink. The divergence of the vector field is $-\sigma - 1 - \beta$, which for the standard parameters is $-13\tfrac23$: a constant, negative, everywhere. So a blob of starting conditions loses volume at a fixed exponential rate, shrinking by a factor of $e^{-13.67}$ per unit time. After a short while its volume is zero for any practical purpose, and in the limit it is exactly zero.

Nearby trajectories separate. The largest Lyapunov exponent is about $+0.9$. Two starting points a millionth apart are a tenth apart within fifteen time units.

Both are true simultaneously, and the resolution is that the three exponents are roughly $(+0.9,\ 0,\ -14.6)$. They sum to the divergence, which is negative — so volume shrinks — while the largest is positive — so distances grow. The blob is stretched along one direction and squashed far harder along another, and the result is a filament of enormous length and no thickness.

That combination is exactly what produces a set with a fractional dimension. The attractor’s box-counting dimension is measured at about $2.06$: more than a surface, less than a solid, and not equal to any integer.

What the two lobes are #

The lobes are not decoration; each is wrapped around a fixed point, and the fixed points have a formula.

Setting all three derivatives to zero gives the origin, and two more at

$$x = y = \pm\sqrt{\beta(\rho - 1)}, \qquad z = \rho - 1.$$

At $\rho = 28$ that is $\pm\sqrt{72} \approx \pm 8.485$, and those are the two dots in the figure. The generator computes them from that expression and evaluates the vector field there, asserting the result is zero to twelve decimal places — a check that would catch a sign error anywhere in the equations.

What the trajectory does is spiral away from each of them. Both are unstable at $\rho = 28$, so an orbit near one spirals outward until it is thrown across to the other, where the same thing happens. Neither is an attractor; together they organise a set that is.

The parameter that decides #

The two-lobe picture is not what the system does at every $\rho$, and a figure that shows it unconditionally is showing a coincidence of parameter choice.

The changeover has a value. The non-zero fixed points are stable up to

$$\rho_c = \frac{\sigma(\sigma + \beta + 3)}{\sigma - \beta - 1} \approx 24.74,$$

and unstable above it. Below $\rho_c$ the system settles; above it, it cannot, and the two-lobe wandering is what happens instead.

This is worth dwelling on because it is a case where a figure’s assertion was written from the famous picture and was wrong. The generator originally demanded two lobes at every parameter, and $\rho = 14$ refused it — correctly. The fix was not to loosen the check but to make it conditional on $\rho_c$, so the assertion now tests the bifurcation rather than the reputation. A guard that only passes at the parameter everyone draws is a guard that has never been tested.

What happens on the way up #

Between the settling case and the standard picture the system passes through a sequence of changes, and following $\rho$ upward makes the two-lobe object look inevitable rather than exotic.

Below $\rho = 1$ the origin is the only fixed point and everything goes to it: no convection, the fluid simply conducts heat.

At $\rho = 1$ the origin loses stability and the two symmetric fixed points appear out of it — a pitchfork bifurcation. The fluid starts rolling, one way or the other, and which way is decided by the initial condition. Two attractors, both simple.

Between $1$ and $\rho_c$ those two points stay attracting, and the approach becomes a spiral as the eigenvalues at each acquire an imaginary part. The $\rho = 14$ figure above is this regime.

At $\rho_c \approx 24.74$ both lose stability in a subcritical Hopf bifurcation, and the qualifier matters: the periodic orbit involved is unstable and shrinks onto the fixed point as $\rho$ rises, so there is nothing stable for the trajectory to move to. The attractor it lands on instead already exists below $\rho_c$ — from about $\rho = 24.06$ there is a strange attractor coexisting with the two stable points, and which one an orbit reaches depends on where it started.

That coexistence region is the interesting part of the story. There is a window of parameters where the same equations have both a tidy answer and a chaotic one, with interleaved basins deciding between them — the arrangement Newton’s method had, arrived at from completely different equations.

Why it is not a cycle #

The obvious objection to the hero figure is that the trajectory might simply have a very long period, and that the picture is a loop drawn too coarsely to see closing.

It is not, and there are three independent reasons.

The exponent is positive. A periodic orbit has largest exponent zero. A measured positive exponent rules out a cycle of any period.

The Poincaré section is a curve, not points. Record where the trajectory crosses a fixed plane. A period-$k$ cycle gives $k$ points; this gives a curve, densely filled, and the map from one crossing to the next is essentially the tent map — the same one-dimensional chaotic map this field started with.

It has been proved. Tucker showed in 2002, by a rigorous computer-assisted argument with interval arithmetic, that the Lorenz system at the standard parameters really does have a strange attractor. That was Smale’s fourteenth problem, posed in 1998, and the answer took four years and a computer.

The method there is worth a sentence, because “computer-assisted” is doing something specific. Interval arithmetic replaces every number with an enclosing interval and every operation with one that produces an interval provably containing the true answer, so a computation returns a bound rather than an estimate. Running the system that way over a covering of the phase space establishes properties of every trajectory in each box at once, not of the particular trajectories sampled. It is a proof in the ordinary sense, carried out by a machine because the case analysis is too large by hand — which puts it beside the four-colour theorem rather than beside a simulation.

Turning the knob further #

Past the standard parameters the system does not simply get more chaotic, which is worth seeing because it is the same pattern the interval maps showed.

Between $\rho = 28$ and $\rho = 350$ there are wide bands of parameters where the attractor is a simple periodic orbit — a closed loop, drawn once, with no strangeness at all — separated by bands where it is chaotic. Around $\rho \approx 100.8$ and again near $\rho \approx 160$ there are stable cycles; above about $\rho = 313$ the system settles into a single large loop and stays there for every larger value.

That is the window structure of the logistic map in a continuous system, and the correspondence is not loose: the windows arrive by period-doubling cascades running backwards, with the same Feigenbaum ratio between successive doublings. A three-dimensional flow and a one-dimensional map, sharing a constant, because both reduce to a unimodal return map.

So “the Lorenz attractor” names a particular parameter value’s answer. The system has a parameter, the parameter has a diagram, and the famous picture is one column of it.

The word “strange” #

Ruelle and Takens introduced the term in 1971, and it is worth stating what it does and does not require.

An attractor is a set that nearby trajectories approach and that contains no smaller such set. A fixed point is one; a limit cycle is one; a torus is one. All three are manifolds — objects with a dimension that is a whole number, and a local structure that looks like ordinary space.

A strange attractor is one whose geometry is not a manifold — not a point, curve or surface, but a set with fractional dimension — and on which the dynamics are sensitive to initial conditions.

The two conditions are separable, and the pairing is a fact rather than a definition. Stretching gives sensitivity, contraction gives zero volume, and the two together force the fractal structure: a set that is stretched and folded back into itself forever has to have layers at every scale, because folding a layer produces two.

Reading the object as a stack of infinitely many sheets is closer to what it is than reading it as a surface. The apparent surface in the figure is a bundle of sheets whose spacing is below any resolution, which is why the dimension comes out slightly above two rather than exactly two.

The $0.06$ is computable rather than measured, which is the satisfying part. Kaplan and Yorke’s formula builds the dimension out of the Lyapunov exponents: take as many as can be added while the running sum stays positive, then add the fraction of the next one that brings the sum to zero. Here $0.9 + 0 = 0.9$, and $0.9/14.6 \approx 0.06$, giving $2.06$. A geometric quantity — how much space the set takes up at small scales — comes out of three separation rates, and agrees with direct box-counting.

That is the same move as measuring a fractal by how it scales rather than by looking at it, and it is the only practical way to get a dimension for an object nobody can draw at more than one scale.

What the drawing is not showing #

The figures here are projections, and a projection of a three-dimensional curve loses the one property that makes the object comprehensible.

The trajectory never crosses itself. In three dimensions, a solution curve of a differential equation cannot intersect its own path — if it did, the state at the crossing would have two different futures, and the equations give exactly one. Every apparent crossing in the figures is two strands at different depths.

That is the whole reason three dimensions are needed. In two dimensions the no-crossing rule is so restrictive that chaos is impossible: a closed curve in the plane separates inside from outside, a trajectory cannot get past its own path, and the Poincaré–Bendixson theorem says the only options left are a fixed point or a limit cycle. Three is the smallest number of dimensions in which a continuous system can be chaotic, and that is a theorem rather than an observation.

The discrete maps of this field escape the restriction by not being continuous in time. An interval map jumps, so nothing stops the orbit landing wherever it likes, and one dimension is enough. The dimension counts are not comparable between the two settings, which is worth remembering when a system is described as “low-dimensional chaos”.

Where this sits #

Two connections close the field.

It is the continuous version of the interval maps. The Poincaré section reduces the three-dimensional flow to a one-dimensional map, and that map is chaotic in the sense the first essays of this field established. The apparatus built there — cobwebs, exponents, stretch-and-fold — is the apparatus that applies here, one dimension down.

It cannot happen in a conservative system. An attractor requires volume contraction, and Poincaré recurrence forbids that in a volume-preserving system. So strange attractors and recurrence are alternatives, and which one a system has is decided by a single sign — the divergence of its vector field.

The other thing to take from the picture is the one Lorenz took. He was not looking for a strange attractor; he was checking a weather model and noticed that a rounded restart diverged from the original run. Everything above is the geometry underneath that observation, and it was found by someone who was looking at something else.