CFD Chaotic Advection: Standard Map Lyapunov Spectrum

Abstract

We begin a computational fluid dynamics conjecture program on bigcompute.science using the same custom-CUDA methodology as our number-theory experiments. The first target is the Chirikov standard map on the 2-torus — an area-preserving map that models chaotic advection in periodically driven 2D flows.

For each coupling parameter $K$, we estimate the maximal Lyapunov exponent $\Lambda(K)$ (largest LCE — standard dynamical-systems terminology) by averaging Benettin tangent-vector growth over thousands of random initial conditions on a single RTX 5090. See the finding claim-validation table for what we do and do not assert about scale.

Why this map?

The standard map

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

is the simplest symplectic model exhibiting a transition from integrability ($K=0$) to widespread chaos ($K \gtrsim 0.972$). In fluid mechanics, identical phase-space structure arises in Stokes flow with periodic forcing — passive tracers mix chaotically even when the velocity field is laminar.

This connects our transfer-operator / ergodic-theory expertise to CFD conjectures without requiring a full Navier–Stokes DNS stack on day one.

Method

1. Grid $K \in [0, K_{\max}]$ with n_k points
2. For each $K$, sample n_ic random $(\theta, p) \in \mathbb{T}^2$
3. Iterate $n_{\mathrm{iters}}$ steps; accumulate $\frac{1}{n}\sum \log\|J v\|$ (Benettin)
4. Output CSV: mean, std, min, max, fraction of ICs with $\Lambda > 0$
5. Certificate: exit code 2 on NaN/Inf; validate $\Lambda(0) \approx 0$

Reproduction

git clone https://github.com/cahlen/idontknow.git
cd idontknow
./scripts/experiments/cfd-chaotic-advection/run.sh 64 512 5000 2.0   # smoke test
./scripts/experiments/cfd-chaotic-advection/run.sh                  # overnight defaults

Requires CUDA 13+, RTX 5090 (-arch=sm_120) or adjust architecture flag.

Dataset: cahlen/cfd-chaotic-advection on Hugging Face — Lyapunov sweeps, certifying logs, and claim-validation artifacts. Finding: Standard Map Chaos Onset.

Next steps

• Compare empirical chaos onset against literature $K_{\mathrm{crit}}$ → finding
• Extend to linked twist maps and sinewave flow models for laminar chaotic mixing
• Phase 2: 2D pseudospectral Navier–Stokes blowup search (Beale–Kato–Majda) → experiment

Human–AI collaboration. Not peer-reviewed. All code open for verification.