danf 0.5.0

Differential Algebra Normal Forms for symplectic maps

DANF

Differential Algebra Normal Forms for symplectic maps, built on DACEyPy.

Computes amplitude-dependent tune shifts (detuning), chromaticity, and the complex-basis normal form of nonlinear symplectic maps via order-by-order elimination in the complex eigenvector basis.

Installation

pip install danf

Quick start

import numpy as np
from daceypy import DA
from danf import NormalForm
from danf.elements import rotation, thin_sext, compose

# Initialize DA engine: order 7, 2 variables (x, px)
DA.init(7, 2)

# Build a one-turn map: rotation (tune 0.31) + sextupole kick (k2L = 1.0)
one_turn = compose(rotation(0.31), thin_sext(1.0))

nf = NormalForm(one_turn)
nf.compute()

print(f"Tune: {nf.tunes[0]:.6f}")
print(f"dnu/dJ: {nf.detuning['dnux_dJx']:.8f}")

API

NormalForm(da_map)

The central class. Accepts a 2-, 4-, or 6-component DA map (1-, 2-, or 3-DOF).

• nf.compute() — run the full computation.
• nf.tunes — linear tunes, one per DOF.
• nf.detuning — amplitude-dependent tune-shift dict.
  • 1-DOF keys: dnux_dJx, alpha_xx, c21.
  • 2-DOF keys: dnux_dJx, dnux_dJy, dnuy_dJx, dnuy_dJy, plus the complex order-3 coefficients c21_xx, c21_xy, c21_yx, c21_yy.
• nf.nf_map — the post-transformation Floquet normal-form map (real-coord DA).
• nf.to_complex_basis(plane=0) — {(k, l): complex} for z = (x - i*px)/√2.
• nf.driving_terms(plane=0, tol=1e-12) — non-Birkhoff (k - l ≠ 1) coefficients only; empty to tolerance away from resonance.

chromaticity_1d(da_map)

Linear and higher-order chromaticity from a 1-DOF map built with one or more passive parameters (e.g. δ as the third DA variable via DA.init(N, 3)).

from daceypy import DA
from danf import chromaticity_1d

DA.init(5, 3)  # 1 DOF + δ as DA(3)
x, px, delta = DA(1), DA(2), DA(3)
mu = 2*np.pi*(0.28 + 1.5*delta - 3.0*delta*delta)
xr  =  x*mu.cos() + px*mu.sin()
axr = -x*mu.sin() + px*mu.cos()

chrom = chromaticity_1d([xr, axr - 0.5*xr*xr])
print(chrom.tune)            # 0.28
print(chrom.dnu_ddelta())    # 1.5
print(chrom.d2nu_ddelta2())  # -6.0

danf.elements

DA-map convenience functions: rotation, thin_quad, thin_sext, thin_oct, drift, compose.

compute_detuning_1d(da_x, da_px, tune)

Low-level 1-D normal-form primitive; returns (dnu_dJ, c21, nf_x, nf_px). Use NormalForm for the standard path.

Cross-validation

DANF's numerical output is regression-tested against COSY INFINITY v10.2 (DANF and TS commands) to 10–12 significant figures on:

Quantity	Agreement
Linear tune, 1-DOF	< 1e-14
dnu/dJ, 1-DOF sextupole map	< 1e-10
dnu_x/dJ_x, dnu_x/dJ_y, 2-DOF	< 1e-10
dnu/ddelta, d²nu/ddelta² (chrom.)	< 1e-10
Birkhoff c_{2,1} complex coeff	< 1e-10

License

MIT