dynsys op• Data kinds: none → signal (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.dynsys_lyapunov_spectrum(system='lorenz', params=None, x0=None, t_end=200.0, dt=0.005, burn_in=20.0) (to call the implementation directly, import mathops; mathops.dynsys_lyapunov_spectrum(system='lorenz', params=None, x0=None, t_end=200.0, dt=0.005, burn_in=20.0); from the registry, opsmath.get("dynsys_lyapunov_spectrum"))
The Lyapunov spectrum by tangent flow + QR — and the sum you can check.
Integrates the state together with an orthonormal frame of tangent vectors (the variational equation dY/dt = J(x) Y), re-orthonormalising by QR at every step and accumulating log of the diagonal. The exponents come out ordered, largest first.
★★Why this earns its place — the trace identity. The sum of the exponents equals the time-average of the divergence of the field:
sum(lambda_i) == <div f>
For Lorenz the divergence is the constant -(sigma + 1 + beta), so the sum is known in closed form: -13.6667 for the classical parameters. That is an exact target the attractor picture cannot provide. The published largest exponent (≈ 0.906 for sigma=10, beta=8/3, rho=28) is a second, independent check.
Returns a signal: the exponents, descending.
Raises ValueError: unknown system; non-finite input; burn_in not shorter than t_end; a trajectory that left float range.
Limits: the exponents converge like 1/sqrt(T) — a short window gives a plausible but wrong spectrum. The trace identity converges much faster and is the honest gate; the individual exponents need long windows.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_what_a_picture_cannot_check — py -3.11 examples/poc_what_a_picture_cannot_check.py
signal as input)mat_solve · mat_lstsq · stat_describe · stat_histogram · stat_zscore · interp_linear · interp_cubic · interp_scattered
dynsys)ode_flow_states · ode_vector_field_grid · dynsys_poincare_section · dynsys_bifurcation_map · dynsys_correlation_dimension
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.