D04 — Lorenz Strange Attractor
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | Lorenz system (σ=10, ρ=28, β=8/3) |
| Group | ℤ₂ (two-lobe symmetry) |
| H^k tier | H¹ |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · LABEL |
| Paper | Paper 512 |
Physical system
The Lorenz system (1963) is a 3-dimensional ODE system:
ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz
At the classical parameter values (σ=10, ρ=28, β=8/3), solutions converge to a strange attractor with fractal dimension d_f ≈ 2.06. The system has a ℤ₂ symmetry (x,y) → (−x,−y) and exhibits sensitive dependence on initial conditions (chaos). The butterfly-shaped attractor is the canonical example of a strange attractor in a dissipative system.
Target category
Vect(ℝ)³ — the category of flows on ℝ³, with morphisms = smooth conjugacies between flows. The attractor A ⊂ ℝ³ is the invariant object; its topology is encoded in the Lorenz template (a branched 2-manifold with a ℤ₂ branch locus).
Interpretation functor
F: C → Vect(ℝ)³ defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Flow on the attractor: trajectories wind around one lobe, then switch — each completed lobe traversal is one ORBIT cycle |
| TWIST | Lobe-switching event: trajectory crosses the branch locus of the Lorenz template; H¹ class changes (left-lobe vs right-lobe winding) |
| LABEL | Symbolic dynamics label: L (left lobe) or R (right lobe) per ORBIT cycle; eigenvalue = Lyapunov exponent λ₁ ≈ +0.906 |
ISA programme
INIT: LABEL[(x₀,y₀,z₀)] -- initial condition near attractor
FLOW: ORBIT[Φ_t(x,y,z)] -- Lorenz flow for time t
SWITCH: TWIST[L ↔ R | z < z_saddle] -- lobe-switch when trajectory crosses branch
SYMBOL: LABEL[s_n ∈ {L,R}] -- record symbolic sequence
LYAP: LABEL[λ₁ = lim(1/t)log|δx(t)/δx(0)|] -- maximal Lyapunov exponent
Computable output
- Lyapunov exponent λ₁ ≈ +0.906 (positive = chaos). This is the eigenvalue of the ORBIT+TWIST programme: the rate at which the TWIST lobe-switching amplifies initial uncertainty.
- Fractal dimension d_f ≈ 2.06 (Kaplan-Yorke formula from Lyapunov spectrum). Non-integer dimension = the attractor spans both H⁰ (two fixed points, unstable) and H¹ (the two-lobe TWIST cycles) without reaching H² (no non-abelian holonomy — the symmetry group is just ℤ₂).
- Symbolic sequence {L,R}^ℕ: the complete topological description of the attractor. Any bi-infinite sequence is realised by some trajectory (the system is topologically conjugate to a subshift of finite type on {L,R}).
- Predictability horizon: t* ≈ (1/λ₁) log(Δ₀/ε) where Δ₀ is initial uncertainty and ε is forecast tolerance. Beyond t*, ORBIT predictions diverge — the symbolic sequence becomes unpredictable.
Why H¹ (not H⁰ or H²)
- Not H⁰: the attractor is not a fixed point or stable limit cycle. ORBIT does not close periodically — trajectories never repeat exactly.
- H¹: the two lobes are the two generators of π₁ of the Lorenz template. The lobe-switching TWIST event is literally a non-contractible loop in the attractor’s topological template. The ℤ₂ symmetry is the H¹ holonomy group.
- Not H²: the symmetry group ℤ₂ is abelian, so no non-abelian BIND is needed. A Rössler attractor (genus-1 template, single lobe, no branching) is H⁰. The Lorenz attractor’s branching makes it H¹. A hypothetical attractor with non-abelian monodromy (e.g., a figure-eight knot complement flow) would be H².
Validation
- λ₁ ≈ +0.906: confirmed to 3 significant figures by Tucker (2002) computer- assisted proof that the Lorenz attractor exists and is chaotic.
- Tucker’s proof is itself an ORBIT fixed-point computation: it constructs a trapping region and verifies that the ORBIT (Lorenz flow) maps it to itself.
- Symbolic dynamics {L,R} coding: confirmed by Guckenheimer & Williams (1979) geometric Lorenz attractor theory.