Principia Orthogona · Book 4 · Chapter 2½ · Between the second and third dimension
Strange attractors: the objects between two and three dimensions
Chapter 1½ looked at curves that fill in between a line and a plane. A chaotic flow in three variables does the same thing one step up: it contracts volume, so its attractor is thinner than a solid, yet it never settles into a curve. Pick a system, drag to rotate, and read the numbers computed for it.
The plate
What was computed
For each run, attractors-verify.py integrates the equations with a fourth-order Runge–Kutta step, discards a transient, and computes the full Lyapunov spectrum with the Benettin method (QR renormalization, tangent map propagated alongside the orbit). The Kaplan–Yorke dimension is then read off the spectrum; it is a conjectured estimate of the attractor's dimension, not a proved value, and any exponent within 0.01 of zero is counted as zero. Two checks guard the code: for systems whose divergence is constant, the exponents must add up to it exactly, and the Burke–Shaw field must be unchanged by the half-turn (x, y, z) → (−x, −y, z).
All constant-divergence rows pass the sum check (the sum column equals the exact divergence column to four decimals). Both symmetry tests pass at 200 random points each. For Lorenz, λ1 = 0.909 and Kaplan–Yorke dimension 2.062, consistent with the values usually quoted for the classical parameters (about 0.906 and 2.06, quoted from memory of the literature and not checked against a source for this chapter).
What the checks changed
Halvorsen at a = 1.89 is not chaotic here. That value is common in pictures, but in my runs λ1 is +0.0015 (indistinguishable from zero) with the others negative: a limit cycle. At a = 1.4 the largest exponent is 0.673 and the system is chaotic. The plate shows both.
Aizawa depends on where you start. With the same parameters, one start gave λ1 = 0.096 (chaotic) and another, the one many write-ups use, gave exponents (0, −, −): a limit cycle. Both are on the plate. A single set of parameters can hold more than one attractor.
Thomas at b = 0.208186 sits at the edge. λ1 there is +0.0023, not distinguishable from zero in this run; at b = 0.18 it is 0.030 and the system is chaotic.
Sprott's case A is conservative. Its exponents sum to about zero (-0.0039), so it has no attractor: its chaotic region is a sea among regular orbits that does not contract. It is on the plate as a contrast, not as a strange attractor.
Sources and limits
Lorenz, Chen and Lü equations and classical parameters: read from arXiv 1409.8649. Their constant divergences, −10 and −19, are my arithmetic from the paper's formula, and the verifier reproduces them.
Aizawa: equations and parameters from this write-up, which is internally inconsistent about the parameter e (it is listed but missing from the written z-equation). I used the form with the e term, as the write-up's own code does.
Burke–Shaw: the parameters s = 10, v = 4.272 and the half-turn symmetry are from Paul Bourke's page, which gives the equations only as an image. The equations I used are the standard form written from memory; the symmetry check agrees with Bourke's statement but does not prove the form is the published one.
Sprott's case A is from this course page. The Rössler, Halvorsen and Thomas equations are the standard forms from memory, not fetched here.
Each exponent is a finite-time estimate from one orbit. They agree between runs to about two decimals where I tried two starts, and no more precision is claimed. This is not a proof of chaos for any system.
This chapter does not claim the systems share an invariant. They share a form (three variables, smooth nonlinear flow) and each has its own computed numbers.
Where this meets the rest of the book
These pages were audited in full on 29 October 2026. What each one is good for, and what to read with care:
Lorenz · Chaos Theory and dm³: the equations, the fixed points, the exponents (+0.906, 0, −14.572) and the Kaplan–Yorke dimension 2.06 agree with this chapter's computed run (+0.9086 and 2.062). Its claim that the Lorenz flow sits in a contact manifold as a Reeb flow is not supported: a Reeb flow preserves volume, and Lorenz contracts it (divergence −13.667, the same number this chapter's verifier reproduces).
μ · The Lyapunov Operator, F · Catastrophe Theory and Disaster Theory: readable accounts of the exponent and of Thom's catastrophes, each now carrying a corrections box listing the statements that do not hold (fold location, catastrophe labels, the n-bonacci rate, the Reeb divergence). Nothing in them is a computed result about the systems on this page.
The Banking Butterfly: a settlement-arithmetic paper that borrows the Lorenz picture as an analogy. It says itself that the mechanism is not chaotic, and it does not use catastrophe theory.
This chapter's role is narrower than those: it adds the other systems and a computed check for each.