Generator

The Lorenz attractor at ρ = 28

A generator in the dynamics library, called 31 times across 8 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

attractor is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates.

`show: "separation"`

A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope.

`show: "section"`

The flow, reduced to one dimension. A scatter of 2395 points: each successive maximum of the Lorenz trajectory's third coordinate against the one before it. The points lie along a single curve with a sharp peak, which is the one-dimensional map the flow induces.

`show: "horseshoe"`

Stretch, fold, and what is left. 5 stages of the horseshoe map's surviving set: one square, then two strips, then four, up to 16, each narrower than the last by a factor of 3.

`show: "henon"`

The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

above the critical parameter the trajectory visits both lobes ×1
and each is narrower by the stretch factor ×1
and even at the cusp, where a straight line cannot fit, the departure stays small ×1
and it is the Lorenz system's own exponent, near nine tenths ×1
and settles on neither fixed point ×1
and the strips are laid out in order without overlapping ×1
areas shrink at every step ×1
below the critical parameter it spirals into a fixed point ×1
between two and four starting gaps, each a whole power of ten from −2 to −12 ×1
enough returns to see a shape ×1
every surviving strip lies inside the square ×1
every window still holds some of the orbit ×1
so the total width falls, and the limit has none ×1
the closed-form fixed points really are fixed ×1
the contraction is between 0.1 and 0.4 ×1
the geometric parameter is between 0.1 and 8 ×1
the growth rate does not depend on how close the two starts were ×1
the Hénon parameter is between 1 and 1.45 ×1
the Jacobian determinant is the contraction ×1
the map's maximum is in its interior, so it folds ×1
the number of magnifications is a whole number between 1 and 4 ×1
the number of stages is a whole number between 2 and 5 ×1
the number of steps is a whole number between 500 and 40000 ×1
the orbit stays bounded, which is what makes it an attractor ×1
the Prandtl parameter is between 0.1 and 40 ×1
the Rayleigh parameter is between 0.1 and 100 ×1
the return map is a curve, not a cloud — its typical thickness is under one per cent of its range ×1
the step size is between 0.0005 and 0.05 ×1
the stretching factor is between 2.2 and 5 ×1
the strips double at every stage ×1
the surviving set has a dimension between nought and one ×1
the trajectory stays in a bounded region ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Dynamics

A closer start buys only time

Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

Dynamics

How fast two orbits part

The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

Dynamics

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

Dynamics

Stretch, fold, and what is left

A system that pushes every pair of nearby points apart and keeps them all inside a bounded region has only one option, and it is the one a baker uses. Stretching and folding is the mechanism, and what survives infinitely many folds is a Cantor set.

Dynamics

The flow that is really a map

A trajectory wandering through three dimensions is hard to reason about. Record only the successive maxima of one coordinate and the wandering collapses onto a curve — a map of an interval to itself, with a corner in the middle, which is a thing the theory can handle.

Dynamics

The orbit a computer draws

A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.

Dynamics

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

Dynamics

Two lobes and no cycle

Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

The whole library · What the figures prove