Worked example: torus dimensions along a Ruelle–Takens–Newhouse route

The torus dimension T^k is read off three independent witnesses: the number of neutral Lyapunov exponents (k zeros: limit cycle 1, 2-torus 2, 3-torus 3 — chaos has a positive one instead), the correlation dimension \(D_2\) of the attractor (\(\approx\) k, fractal for chaos) and of the plane-crossing Poincaré section (\(\approx\) k-1: dots \(\to\) loop \(\to\) band). Measured on Lorenz-96 (Nx=10), the route is textbook RTN, with locking windows interleaving near the breakdown and no stable 3-torus (as RTN predicts — \(T^3\) is generically unstable and the attractor turns strange directly):

F	\(\lambda\) signature (\(n_0\) = neutrals)	\(D_2\) state / section	regime
2–3.95	one zero	1.05 / —	limit cycle (\(T^1\))
4.0–4.06	one zero	—	locked windows (periodic on the torus)
4.2–4.4	two zeros	1.8 / 0.9	quasiperiodic (\(T^2\))
4.45	\(\lambda_1 = 0.016\), \(n_0\)=2	2.0 / 1.0	first chaos, interleaved with...
4.5–4.55	two zeros	2.1–2.2 / 1.0	...re-locked/QP windows (Arnold tongues)
4.6	\(\lambda_1 = 0.039\) (converged)	2.15 / 1.02	chaotic wrinkled torus — section still loop-like
5	\(\lambda_1 = 0.07\) (plateau, \(\lambda_1 T\) grows)	2.9 / 1.1	chaos on a thickened torus remnant
8	3 positive exponents	4.8 / 1.8	developed chaos (\(D_\mathrm{KY} \approx 6.5\))

This is why F=4.6 and F=5 "look QP": the strange attractor inherits the torus geometry (\(D_2\) barely above 2, section barely above a loop) while the dynamics on it are already exponentially divergent — the spectrum, not the geometry, makes the call.

The route in the 8-panel diagnostics

Generated with python -m ntsa.characterize --model lorenz96 --param F --values ... (Nx = 10). On the torus side of the route, the delay portrait fills a surface, the Poincaré section closes into a loop, and the spectrum shows two neutral exponents; past breakdown the section stays loop-like long after \(\lambda_1\) turns positive.

F = 2 \(\to\) 4.06 — limit cycles and locking. Period-1 orbits (\(D_\mathrm{KY}\) = 1, single return-map cluster), then a period-3 window at F = 4 and a locked state at F = 4.06 (\(f_2/f_1 \approx\) 1/2, \(D_\mathrm{KY}\) = 1.43):

F = 4.3 \(\to\) 8 — torus, breakdown, developed chaos. The 2-torus at F = 4.3 (quasiperiodic, \(D_\mathrm{KY}\) = 1.94, loop-shaped section), the wrinkled-torus chaos at F = 4.6 (\(\lambda_1\) = 0.040) and F = 5 (\(\lambda_1\) = 0.083) where the section still looks like a thickened loop, and developed chaos at F = 8 (three positive exponents, \(D_\mathrm{KY}\) = 6.51):