Standard Map Chaos Onset: Λ(K) Crosses Literature K_crit on RTX 5090

The Finding

We computed the maximal Lyapunov exponent (largest Lyapunov characteristic exponent; standard terminology, not a claim of world-record numerical values) $\Lambda(K)$ of the Chirikov standard map — the canonical phase-space model for chaotic advection in 2D incompressible flows — using a custom CUDA kernel on a single RTX 5090.

Deep certifying sweep:

Parameter	Value
$K$ grid	2048 points in $[0, 5]$
Initial conditions per $K$	8192
Iterations	50,000 (Benettin)
Total trajectories	16,777,216
Wall time	116.6 s
NaN/Inf	0

At the literature chaos threshold $K_{\mathrm{crit}} \approx 0.971635406$ (Chirikov, 1979):

$\bar{\Lambda}(K_{\mathrm{crit}}) \approx 0.0446, \qquad \text{fraction}(\Lambda > 0) > 99.9\%$

At $K = 5$: $\bar{\Lambda} \approx 0.957$, compared below to the large-$K$ estimate $\Lambda \approx \ln(K/2) \approx 0.916$ (Chirikov 1979; Cary et al. 1986).

Method details

Initial conditions. For each $K$ on a uniform grid of 2048 points in $[0, K_{\max}]$, we draw 8192 independent uniform random pairs $(\theta_0, p_0) \in [0, 2\pi)^2$ using a per-thread SplitMix64 PRNG seeded by (global_seed, k_index, ic_index) (standard_map_lyapunov.cu).

Lyapunov estimate. We apply the Benettin tangent-vector algorithm (Benettin et al. 1980): at each of 50,000 map steps we multiply a unit tangent vector by the Jacobian, renormalize every iteration, and accumulate $\frac{1}{N}\sum \log\|J v\|$. This yields one finite-time maximal Lyapunov exponent per IC; we report the ensemble mean, standard deviation, per-$K$ min/max over ICs (spread of finite-time estimates, not global records), and fraction with $\Lambda > 0$.

Hardware. NVIDIA GeForce RTX 5090 (32 GB, Blackwell architecture, compute capability 12.0). We compile with -arch=sm_120 under CUDA 13.0 (NVIDIA’s flag for CC 12.0 devices). One CUDA thread per $(K, \mathrm{IC})$ pair; all arithmetic is fp64 in the Benettin loop. The kernel exits with code 2 on any NaN/Inf in tangent norms (certifying run reported zero failures).

Peer review. Three AI audits (gpt-4.1, o3-pro, gemini-2.5-pro) on 2026-05-31; review JSONs and remediations.

External validation

Check	Literature / theory	This sweep
$\bar{\Lambda}(0)$	$0$ (integrable rotation)	$0.000$ exactly
$\bar{\Lambda}(K_{\mathrm{crit}})$	$\approx 0.03$–$0.06$ (Greene 1979; Lichtenberg & Lieberman 1992, Fig. 7.5)	$0.0446$ at $K = 0.9722$
frac($\Lambda > 0$) at $K_{\mathrm{crit}}$	Majority positive in chaotic sea	99.91% (8192 ICs)
$\bar{\Lambda}(5)$	$\ln(5/2) \approx 0.916$ (Chirikov 1979; Cary et al. 1986)	$0.957$ (+4.4%)
Deep vs standard sweep	Qualitative agreement	512-run and 2048-run curves match at shared $K$

The $K=5$ value sits within 5% of the asymptotic $\ln(K/2)$ formula; the small excess is consistent with finite-$K$ corrections documented by Manos & Robnik (2013). A dedicated convergence study (validate_claims.py) shows the $K=5$ ensemble mean is stable to $\sim 10^{-4}$ between 5,000 and 100,000 iterations (65,536 ICs, GPU).

Claim validation (what “largest” does and does not mean)

Claim	Status	Evidence
Maximal LCE via Benettin	Valid standard method	Benettin et al. 1980; Sprott/Wolf numerical guides; symplectic pairing $\lambda_1+\lambda_2 \approx 0$ verified to $10^{-15}$ (CPU, 200 ICs)
Numerical values at $K_{\mathrm{crit}}$, $K=5$	Reproduce literature ranges	Mean $0.0446$ in Greene/Lichtenberg band $0.03$–$0.06$; $K=5$ mean $0.957$ vs $\ln(2.5)=0.916$ (+4.4%)
16.8M trajectories	Large single-GPU parameter sweep	Not the largest standard-map study ever (Chirikov & Shepelyansky 1984; StdMap at dynamical-systems.org; GPU packages Chaoticus 2025, Julia ChaosTools)
max_lyapunov CSV column	Per-$K$ max over 8192 ICs	e.g. $0.998$ at $K=5$ is ensemble spread of finite-time estimates, not a published record
50,000 iterations sufficient	Yes at large $K$; marginal at $K_{\mathrm{crit}}$	GPU: $K=5$ mean stable 5k–100k iters; CPU: $K_{\mathrm{crit}}$ mean drops $\sim 2.5\%$ from 50k to 100k iters (sticky/near-integrable orbits)

We do not claim: world-record computation size, a refined $K_{\mathrm{crit}}$, or fully saturated Lyapunov exponents for every IC at every $K$.

python3 scripts/experiments/cfd-chaotic-advection/validate_claims.py

Data: Hugging Face dataset (deep_sweep, validation configs) · Experiment

Important nuance (read before interpreting the curve)

Finite-time sensitivity is not global chaos. Our Benettin estimate uses 50,000 iterations per initial condition. At small $K$, some individual ICs can show slightly negative finite-time $\Lambda$ (regular islands, slow convergence) even while the mean over 8192 ICs is positive. The heuristic “onset” where $\bar{\Lambda} > 0.01$ appears near $K \approx 0.75$ is therefore not the literature chaos threshold $K_{\mathrm{crit}} \approx 0.972$ — it marks where finite-time tangent growth becomes detectable in our sampling, not where the phase space is globally chaotic.

Mean $\bar{\Lambda}(K)$ is not strictly monotonic. The deep sweep shows tiny non-monotonic wiggles in $\bar{\Lambda}(K)$ at the grid level (IC sampling noise at fixed iteration count). The overall trend is increasing; we do not claim a theorem of monotonicity.

What we do claim: at the established $K_{\mathrm{crit}}$, our certifying sweep finds $\bar{\Lambda} \approx 0.045$ with $>99.9\%$ of ICs positive — consistent with the known integrability-to-chaos transition, not a new threshold estimate.

Why This Matters for CFD

The standard map

$p' = p + K\sin\theta, \qquad \theta' = \theta + p' \pmod{2\pi}$

is area-preserving on $\mathbb{T}^2$. The same structure appears in Stokes flow with periodic forcing: passive tracers can mix chaotically even when the velocity field is laminar (Aref 1984; Ottino 1989).

This experiment is the first published entry in bigcompute.science’s CFD program: a certified GPU Lyapunov sweep with open data and multi-model AI audit. Large standard-map Lyapunov computations exist in the literature (Chirikov & Shepelyansky 1984; Manos & Robnik 2013); our contribution is the reproducible open pipeline (custom CUDA, certifying logs, Hugging Face dataset), not a new numerical value for $K_{\mathrm{crit}}$.

The map connects conceptually to our Hausdorff / transfer-operator work: both study ergodic properties of composition operators on phase space. Here the digit alphabet is replaced by a physical coupling parameter $K$.

Key Results

Integrable limit validated

$\bar{\Lambda}(0) = 0$ exactly (within floating point), as expected for $K=0$ (pure rotation on the torus).

Growth of $\bar{\Lambda}(K)$

$K$	$\bar{\Lambda}(K)$	frac($\Lambda>0$)
0.75	0.011	99.9%
0.972 (literature $K_{\mathrm{crit}}$)	0.045	>99.9%
1.0	0.050	>99.9%
2.0	0.332	100%
5.0	0.957	99.98%

$\bar{\Lambda}(K)$ generally increases with $K$ on our grid; small non-monotonic wiggles appear from finite IC sampling (see nuance above). The transition is gradual, not a sharp step — consistent with a progressive invasion of chaotic orbits near $K_{\mathrm{crit}}$ rather than an instantaneous flip.

Reproducibility

git clone https://github.com/cahlen/idontknow.git
cd idontknow
./scripts/experiments/cfd-chaotic-advection/run.sh 2048 8192 50000 5.0
python3 scripts/experiments/cfd-chaotic-advection/plot_lyapunov.py \
  scripts/experiments/cfd-chaotic-advection/results/lyapunov_k2048_ic8192_iter50000.csv \
  -o lyapunov_spectrum.svg

Dataset: cahlen/cfd-chaotic-advection — Lyapunov sweeps, certifying logs, and validation artifacts.

Limitations (to our knowledge)

• We report the maximal (largest) Lyapunov exponent only — standard for 2D maps. For this area-preserving symplectic map the second exponent is $- \Lambda$ (Liouville; validated numerically to machine precision in validate_claims.py).
• 50,000 iterations appear saturated at large $K$ but may underestimate the ensemble mean slightly near $K_{\mathrm{crit}}$ where sticky orbits converge slowly; per-IC negative finite-time $\Lambda$ values can occur (e.g. min $\Lambda = -4.1 \times 10^{-5}$ at $K_{\mathrm{crit}}$) while ensemble means are positive.
• We do not refine $K_{\mathrm{crit}}$; we compare against Greene’s value $K_{\mathrm{crit}} \approx 0.971635406$ at our grid resolution ($K = 0.9722$ nearest grid point).
• We do not claim the largest standard-map computation in the literature — only an open, certifying single-GPU sweep at this $(K, \mathrm{IC}, \mathrm{iter})$ resolution.
• AI peer-reviewed (not journal peer-reviewed). See verifications.

References

• Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica 15, 9–20.
• Chirikov, B. V. (1979). A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379.
• Greene, J. M. (1979). A method for determining the stochastic transition. J. Math. Phys. 20, 1183–1201.
• Chirikov, B. V., Shepelyansky, D. L. (1984). Correlations and diffusion of chaos in nonlinear systems. Phys. Rev. A 33, 2667–2675.
• MacKay, R. S. (1983). A renormalisation approach to invariant circles in area-preserving maps. Physica D 7, 283–300.
• Cary, J. R., Escande, D. F., Tennyson, J. L. (1986). Adiabatic invariant change due to separatrix crossing. Physica A 13, 475–482.
• Wolf, A., Swift, J. B., Swinney, H. L., Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D 16, 285–317.
• Lichtenberg, A. J., Lieberman, M. A. (1992). Regular and Chaotic Dynamics (2nd ed.). Springer.
• Meiss, J. D. (1992). Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64, 795–848.
• Manos, T., Robnik, M. (2013). The standard map: from the pendulum to the accelerator and beyond. Chaos 23, 013127.
• Cristadoro, G., Maldarella, D., Turchetti, G. (2008). Instability of the periodic motion of a particle in a weakly nonlinear potential. Chaos 18, 013137.
• Aref, H. (1984). Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.
• Ottino, J. M. (1989). The Kinematics of Mixing. Cambridge University Press.

Human–AI collaboration. Code: idontknow/cfd-chaotic-advection.