Reeb versus Legendrian: what the attractor is
The contact form carried through Book IV is α = dz − r²dθ on M = ℝ²+ × ℝ, with dα = −2r dr ∧ dθ. The Reeb field is the unique R with ιRdα = 0 and α(R) = 1. Writing R = a ∂r + b ∂θ + c ∂z:
α(R) = c − r²b = c = 1 ⇒ R = ∂z, uniquely
The dm³ attractor is γ*(t) = (1, t, t) in (r, θ, z). Its tangent is γ̇ = (0, 1, 1), and
So γ* lies in ker α. It is Legendrian. The Reeb field is transverse to ker α and a Legendrian curve lies inside it; a curve cannot be both, and the two conditions — α(γ̇) = 1 against α(γ̇) = 0 — are complementary rather than merely different.
Chapter 3 carries this already: its text reads “It is Legendrian, not Reeb”, with the tangent and the pairing written out. What is added here is not the correction but the computation behind it. Block [1] solves for R symbolically from the two defining conditions and then evaluates α(γ̇), rather than asserting either.
The same block records a second thing, easily lost. z(t) = t is unbounded, so γ* is a helix in M and not a closed curve. T* = 2π is the period of the (r, θ) reduction, not of γ*. Every period quoted in this chapter is that one, and no statement below is a statement about a closed orbit in M.
The theorem
Take a planar system in polar form with the angular speed normalised:
A limit cycle is a positive root r* of f. Along it the transverse variational equation is d(δr)/dt = f′(r*) δr, integrated over T = 2π, so the Floquet multiplier is m = exp(2π f′(r*)). The cycle is non-hyperbolic exactly when m = 1, which is exactly when f′(r*) = 0 — and since r* is already a root, exactly when r* is a multiple root of f.
For the family ṙ = f(r; λ), θ̇ = 1 with f polynomial in r, the bifurcation set in parameter space is { λ : discr f(·; λ) = 0 }, and the Floquet multiplier equals 1 at every point of it.
The discriminant of a polynomial vanishes precisely when it has a multiple root; a multiple root of f is precisely a common root of f and f′; a common root is precisely a cycle with f′(r*) = 0, hence m = e0 = 1. Each step is an equivalence, so the three descriptions cut out the same set. □
Nothing dynamical survives in the statement. The left-hand side is defined by the behaviour of a flow; the right-hand side is computed by elimination in exact arithmetic, from the coefficients, without integrating the flow once.
The dm³ field, deformed inside its own family
The dm³ radial field is ṙ = r(1 − r²), whose positive root is r* = 1. Deform it by the next odd term, so the deformation stays inside the family the original belongs to:
u = r², f = r g(u), g(u) = 1 − u + εu²
discu(g) = 1 − 4ε block [3]
The discriminant is linear. It vanishes at εc = 1/4 and nowhere else, the double root there is u* = 2, and so the fold sits at r* = √2. The multiplier check is immediate: f′(r) = 1 − 3r² + 5εr⁴, and at (r, ε) = (√2, 1/4),
m = exp(2π · 0) = 1 exactly, as Theorem 1 requires
Two cycles exist for 0 < ε < 1/4 — the stable one continuing the dm³ cycle from r = 1, and an unstable one descending from infinity — and they annihilate at the fold. Above εc there is no positive root at all: g has no real roots, and every trajectory escapes.
Numerics: the cycles and the multiplier
The algebraic claims above are checked against integration. Block [4] runs the radial equation from r(0) = 0.5 to t = 400 at rtol 10−12 and compares the endpoint with √u−:
| ε | r* by integration | r* algebraic | |error| |
|---|---|---|---|
| 0.000 | 1.000000000000 | 1.000000000 | 1.92 × 10−13 |
| 0.100 | 1.061610405842 | 1.061610406 | 1.59 × 10−13 |
| 0.200 | 1.175570504585 | 1.175570505 | 9.75 × 10−14 |
| 0.240 | 1.290994448735 | 1.290994449 | 5.14 × 10−13 |
| 0.249 | 1.371507574057 | 1.371507574 | 1.43 × 10−13 |
Block [5] does the same for the multiplier, integrating the variational equation over one period T = 2π and comparing with exp(2π f′(r*)):
| ε | r* | m by variational integration | exp(2π f′(r*)) |
|---|---|---|---|
| 0.0000 | 1.000000000 | 3.4873423564 × 10−6 | 3.4873423562 × 10−6 |
| 0.2000 | 1.175570505 | 4.2371726419 × 10−4 | 4.2371726419 × 10−4 |
| 0.2400 | 1.290994449 | 1.5164619865 × 10−2 | 1.5164619865 × 10−2 |
| 0.2500 | 1.414213562 | 1.0000000000 | 1.0000000000 |
The last row is the theorem. The multiplier climbs across four orders of magnitude as ε approaches the fold — from e−4π at the dm³ cycle, where contraction is fierce, to 1 — and the exact place it arrives at 1 was known before the first integration was run.
εc by search, εc by discriminant
Block [6] does it the other way: brackets the fold between ε = 0.20, where a cycle exists, and ε = 0.30, where none does, and bisects sixty times, deciding at each step by integrating to t = 300 and asking whether the trajectory settled on a finite root.
εc = 1/4 exactly: 0.250000000000000 gap 3.40 × 10−6
Sixty bisection steps, each one an initial-value problem integrated to t = 300, and the answer is correct to six figures. The discriminant is correct to all of them, and it is a rational number, and it took one resultant. The gap is not an indictment of the search — near a fold the surviving cycle’s basin narrows and its approach slows, so any decision rule based on a finite horizon degrades exactly where the answer matters most. That is a property of the method, not of the integrator.
This is the chapter in one line. The bifurcation set was never a thing to be found by looking. It is the zero set of a polynomial in the coefficients, and it was available before the model was integrated once.
The e−z coupling, and what the reduction is
Everything above is planar. System (3.1) is not: it carries a coupling that ties the radial equation to z.
θ̇ = 1
ż = r² − 2(r − 1)²e−z
Block [7] integrates the full three-dimensional system from (0.9, 0, 0) to t = 400 and compares the radius reached with the planar algebraic root:
| ε | 3-D r(400) | planar r* | |difference| | z(400) |
|---|---|---|---|---|
| 0.000 | 1.000000000000 | 1.000000000000 | 4.49 × 10−13 | 400 |
| 0.200 | 1.175570504586 | 1.175570504585 | 1.05 × 10−12 | 552 |
| 0.240 | 1.290994448736 | 1.290994448736 | 3.96 × 10−13 | 666 |
The coupling decays like e−z, and z grows without bound on the attractor. So the planar reduction is asymptotic: exact in the limit z → ∞, and not at any finite z. The table is a measurement at one horizon, not a bound. What it licenses is the statement that the three-dimensional system inherits the planar bifurcation set in the limit; it does not license an error estimate at finite time, and none is proved here.
The basin figure, and how much of it may be quoted
Block [8] regenerates the inner basin boundary at z = 0 for ε = 0 — the figure printed as r* ≈ 0.776, refined in Chapter 10 — by bisecting on whether the trajectory converges to the attractor. It is a plain float bisection driven by an integrator at rtol 10−11, atol 10−13, so its error floor is around 10−13 and digits past the twelfth are integrator noise. The rigorous value lives in certify_rstar_rigorous.py, which brackets it by mpmath centre, Jacobian-linearised error transport, and an interval-Hessian Lagrange remainder:
block [8] : 0.775940575502539698 above the upper bound by 1.96 × 10−13
The float value lies outside the certified bracket. It is not inside it and the check does not pretend otherwise: the twelve leading digits are identical — 775940575502 — and the two differ at the thirteenth, by an amount comparable to the width of the bracket itself. Eleven significant figures agree under rounding; that is the whole of what may be claimed.
Block [8] passes, and what it asserts is worth stating exactly, because a stronger assertion was available and would have been false. It checks that the float bisection sits within its own error floor of the certificate, and that the first eleven significant figures match. A check demanding that the float land inside the rigorous bracket would fail — and it would be the check that was wrong, not the number.
Chapter 3 quotes ≈ 0.776 and Chapter 10 establishes it. Both are safe at that precision. Anything past eleven significant figures must be cited from the certificate and not from this block.
The same curve, one chapter earlier
Block [9] closes the loop with Chapter 23. The discriminant of the depressed cubic and the bifurcation set of the quartic potential are the same curve:
V(x) = x⁴/4 + ax²/2 + bx, Res(V′, V″) = 4a³ + 27b² the cusp catastrophe
The sign is a convention on the discriminant and does not move the zero locus. Chapter 23 obtained 4a³ + 27b²c as the dual of the cuspidal cubic, and 4a³ + 27b² in the chart c = 1; its block [12] noted the coincidence and pointed here. The elimination performed there on a family of plane curves and the elimination performed here on a family of vector fields are the same elimination. Dual variety, discriminant, bifurcation set: what changes between the three names is what the parameters are called, and nothing else.
That is the sense in which the ladder has become algebraic. A question posed about the long-run behaviour of a flow has been answered by a resultant, and the answer is a rational number rather than a converged search.
Honest inventory
| Claim | Status | What it rests on |
|---|---|---|
| Reeb field of α = dz − r²dθ is ∂z | Computed | block [1], solved from the two defining conditions |
| the dm³ attractor is Legendrian, not a Reeb orbit | Computed | block [1]; α(γ̇) = 1 − r² = 0 at r = 1 |
| γ* is a helix; T* = 2π is the reduction’s period | Computed | block [1]; z(t) = t is unbounded |
| bifurcation set = { discr f = 0 }, multiplier 1 on it | Proved | §24.2; three equivalences, elementary |
| m = exp(2π f′(r*)) | Classical | Floquet theory for the transverse variational equation |
| discu(1 − u + εu²) = 1 − 4ε | Computed | block [3] |
| εc = 1/4, u* = 2, r* = √2 | Computed | block [3], exact rational |
| f′(√2; 1/4) = 0, so m = 1 exactly at the fold | Computed | block [3] |
| cycle radii agree with the roots to 10−9 | Computed | block [4], five values of ε |
| multipliers agree to 10−6 relative | Computed | block [5], four values of ε |
| bisection lands 3.40 × 10−6 from 1/4 | Computed | block [6], 60 steps |
| the 3-D system inherits the planar root at t = 400 | Computed | block [7]; a horizon, not a bound |
| the planar reduction is exact at finite z | FALSE | coupling is e−z; exact only as z → ∞ |
| a finite-z error bound for the reduction | OPEN | not attempted here; block [7] measures, it does not bound |
| basin boundary ≈ 0.776 at z = 0 | Computed | block [8]; Ch 10 refines, Ch 3 quotes |
| block [8]’s float value lies inside the certified bracket | FALSE | above it by 1.96 × 10−13; eleven figures is the limit |
| 4a³ + 27b² is dual, discriminant and bifurcation set | Computed | block [9]; Ch 23 block [12] |
| the dm³ flow algebraises | OPEN | carried from Ch 23: the helix z = θ is transcendental in these coordinates |
Every row marked Computed is regenerated by ch24-verify.py, which sits beside this chapter and exits non-zero on any failure. Two rows are marked FALSE deliberately. They are claims a reader might reasonably assume the chapter is making, and it is not: the planar reduction is a limit, and the basin figure has eleven honest significant figures rather than eighteen. A verification script that reported neither would still exit zero, and would be worth less for it.