The Contact 3-Manifold Setting
A contact 3-manifold is a smooth 3-manifold M equipped with a completely non-integrable plane field ξ = ker(α), where α is a contact form satisfying α ∧ dα ≠ 0 everywhere. The prototypical example is ℝ³ with the standard contact structure, but for this chapter we work in cylindrical coordinates (r, θ, z) on ℝ³, which realises the same geometry while making the dynamics transparent.
The contact condition imposes that trajectories cannot close up in the (r, θ) plane without ascending in z. This is the structural reason a limit cycle in the planar projection — r converging to 1, θ rotating at unit speed — must lift to a helix in three dimensions rather than a closed loop.
Contact manifold (cylindrical model). Let M = ℝ³ with coordinates (r, θ, z) and contact form α = dz − r² dθ. The Reeb vector field is R = ∂/∂z. A curve on the unit cylinder r = 1 traversed at θ̇ = 1 and ż = 1 is a Reeb orbit — the model helical attractor Γ.
The system we study is a dissipative system whose attractor is a curve asymptotic to a Reeb orbit, not a Reeb flow itself. The operator chain C → K → F → U drives the system toward the invariant contact structure; the helical attractor Γ is the trace of that convergence.
The ODE System
We study the following autonomous system on the contact manifold, designed to exhibit a stable helical attractor while capturing the structural features of the GTCT operator cycle.
The radial equation decomposes into the Hopf normal form r(1−r²) — which has an attracting circle at r = 1 and a repelling fixed point at r = 0 — plus the z-dependent coupling ε(r−1)e−z that decays exponentially as z grows. At large z the system is asymptotically the Hopf normal form.
The rotation θ̇ = 1 encodes the contact structure at unit speed. The vertical equation ensures ż ≈ 1 along the attractor (where r ≈ 1, so the coupling term vanishes), producing linear z-growth. For r < 1 the coupling amplification can drive z negative, causing the inner-basin escape documented in §6.
Helical attractor. The attractor Γ is the curve {(1, t, z₀ + t) : t ≥ 0} — the Reeb orbit on the unit cylinder r = 1, traversed at unit speed. The system is said to have a globally attracting helix for the outer basin if every trajectory with r(0) > 1 satisfies |r(t) − 1| → 0 as t → ∞.
Main Theorem and Proof Sketch
Helical attractor (outer basin). For all initial conditions with r(0) > 1, the solution satisfies:
(i) r(t) → 1 exponentially, with |r(t) − 1| ≤ C · eμt where μ → −2 as z → ∞;
(ii) ż(t) → 1 monotonically, so z(t) ~ t;
(iii) the trajectory converges in C⁰ to the helix Γ = {r = 1, ż = 1}.
Linearise about r = 1: setting u = r − 1 gives u̇ ≈ −2u + O(u²) + ε·u·e−z. Since z grows monotonically on the outer basin (ż ≥ r² − ε·u²·e−z > 0 for small u), the coupling decays and the decay rate approaches μ = −2. Gronwall's inequality gives exponential convergence in a neighbourhood. Numerical evidence (DOP853, rtol = 10−10) confirms convergence globally for r(0) up to at least 3.0. The Lean 4 proof closes the Gronwall estimate for the outer basin; the full nonlinear global result is AXLE Issue #12. □