D05 — KAM Tori and Chaos Onset
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | Nearly-integrable Hamiltonian systems |
| Group | 𝕋ⁿ (n-torus, action-angle symmetry) |
| H^k tier | H⁰ / H¹ / H² |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND · LABEL |
| Paper | Paper 512 |
Physical system
A nearly-integrable Hamiltonian H(I, θ) = H₀(I) + ε H₁(I, θ), where (I, θ) are action-angle variables on ℝⁿ × 𝕋ⁿ and ε ≪ 1 is the perturbation strength. The KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962) guarantees that most invariant tori of H₀ survive the perturbation, provided the frequency vector ω(I) = ∂H₀/∂I satisfies a Diophantine non-resonance condition. As ε increases, resonant tori break first (Poincaré-Birkhoff theorem), then the last KAM torus at ε = ε* (Greene’s criterion), beyond which global chaos appears.
Target category
Symp — the category of symplectic manifolds, with objects = Liouville- integrable systems (𝕋ⁿ fibrations over ℝⁿ) and morphisms = symplectic conjugacies. The H^k grading is by the fate of each torus under perturbation.
Interpretation functor
F: C → Symp defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Quasi-periodic motion on a KAM torus: the flow Φ_t winds around 𝕋ⁿ with frequency vector ω. Closed (H⁰) iff ω is rationally dependent; dense (H¹) iff Diophantine |
| TWIST | Resonance: ω·k = 0 for some k ∈ ℤⁿ; the torus breaks into a chain of islands (Poincaré-Birkhoff fixed points) with a non-trivial H¹ holonomy |
| BIND | Last KAM torus destruction at ε*: the H² obstruction class (cantorus) forms — a Cantor set remnant of the broken torus with non-zero flux |
| LABEL | Action variable I: eigenvalue of ORBIT = winding number ratio ω₁/ω₂ (rotation number) |
ISA programme
INTEGR: ORBIT[Φ_t on 𝕋ⁿ | ε=0] -- unperturbed quasi-periodic orbit (H⁰)
PERTURB: TWIST[H₁(I,θ) | ε small] -- resonance zones open (H¹ islands)
THRESH: LABEL[ε* | Greene residue = 1/4] -- critical perturbation (β* snap)
CANTORUS: BIND[cantorus flux Φ_flux] -- last torus breaks → H² obstruction
CHAOS: ORBIT fails to close -- global diffusion above ε*
Computable output
- Rotation number ρ = ω₁/ω₂: rational ρ = p/q → periodic orbit (H⁰); irrational Diophantine ρ → KAM torus (H⁰ dense); noble number ρ = (√5−1)/2 (golden mean) → last surviving torus.
- Critical perturbation ε: computed by Greene’s residue criterion — the residue R of the period-q approximants to the last torus converges to 1/4 at ε = ε. Computable to arbitrary precision.
- Cantorus flux Φ_flux(ε > ε): the H² obstruction class surviving after the last torus breaks. Quantifies the rate of Arnold diffusion through the broken torus. Φ_flux = 0 at ε = ε (torus just breaks), grows as (ε − ε*)^α for some exponent α.
- The H⁰/H¹/H² ladder is exact:
| Stratum | ε range | Physical state | ISA |
|---|---|---|---|
| H⁰ | 0 ≤ ε ≪ ε* | Integrable tori survive | ORBIT closes |
| H¹ | ε near resonances | Island chains (Poincaré-Birkhoff) | TWIST |
| H² | ε ≥ ε* | Last torus breaks → cantorus | BIND (H² obstruction) |
Validation
- KAM theorem: Kolmogorov (1954), Arnold (1963), Moser (1962) — rigorous existence proof for Diophantine tori.
- Greene’s residue criterion: ε* computed numerically for the standard map (Chirikov-Taylor map) to 10 significant figures. The critical golden-mean torus breaks at K* ≈ 0.971635 (K = ε in standard map notation).
- Cantorus flux: MacKay, Meiss & Percival (1984) confirmed Φ_flux ∝ (K−K*)^α with α ≈ 3.01 for the golden-mean cantorus.
- β* correspondence: ε* is exactly the β* snap event in the ISA — the critical inverse temperature at which the Gibbs distribution over orbits crystallises from a KAM torus (ordered, H⁰) to a cantorus (H² obstruction) to global chaos. The KAM theorem is a finite-temperature (finite-β) stability result: tori survive for β > β*.