nafflib

P. Belanger, K. Paraschou, G. Sterbini, G. Iadarola et al.

A Python implementation of the Numerical Analysis of Fundamental Frequencies algorithm (NAFF [1-2]) from J. Laskar. This implementation uses a tailor-made optimizer (nafflib.optimise.newton_method, from A. Bazzani, R. Bartolini & F. Schmidt) to find the frequencies up to machine precision for tracking data. A Hann window is used to help with the convergence (nafflib.windowing.hann).

An insightful description of the NAFF algorithm is provided in the textbook by A. Wolski [3].

[1] J. Laskar, Introduction to Frequency Map Analysis. http://link.springer.com/10.1007/978-94-011-4673-9_13
[2] J. Laskar et al., The Measure of Chaos by the Numerical Analysis of the Fundamental Frequencies. Application to the Standard Mapping. https://doi.org/10.1016/0167-2789(92)90028-L
[3] A. Wolski,Beam Dynamics in High Energy Particle Accelerators, Section 11.5: A Numerical Method: Frequency Map Analysis. https://www.worldscientific.com/doi/abs/10.1142/9781783262786_0011

Installation

pip install nafflib

Usage

Examples can be found in the examples folder. The harmonics of the data are computed using position-momentum data and the order of the window can be specified by the user. Altough the algorithm works with position-only data, the use of position-momentum is preferred if possible.

Tune

The tune of a signal can be obtained from real or complexe signals as suchs:

# Let's assume the following signal
z = x - 1j * px

# The two following calls are equivalent
# --------------------------------------------------
Q = nafflib.tune(z, window_order=2, window_type="hann")
Q = nafflib.tune(x, px, window_order=2, window_type="hann")
# --------------------------------------------------

# Using the position only:
# --------------------------------------------------
Q = nafflib.tune(x, window_order=2, window_type="hann")
# --------------------------------------------------

Harmonics

Phase space trajectories (x,px),(y,py),(zeta,pzeta) are used to extract the spectral lines of the signal plane-by-plane with the harmonics() function. The number of harmonics is specified with the num_harmonics argument. Again, the function can be used with position only or position-momentum (preferred) information.

# Let's assume the following signal
z = x - 1j * px

# The two following calls are equivalent
# --------------------------------------------------
spectrum = nafflib.harmonics(z, num_harmonics=5, window_order=2, window_type="hann")
spectrum = nafflib.harmonics(x, px, num_harmonics=5, window_order=2, window_type="hann")
# -> where spectrum = (amplitudes,frequencies)
# --------------------------------------------------

# From position only:
# --------------------------------------------------
spectrum = nafflib.harmonics(x, num_harmonics=5, window_order=2, window_type="hann")
# -> where spectrum = (amplitudes,frequencies)
# --------------------------------------------------

Categorization of harmonics

For stable motion sufficiently close to integrable invariants of a conservative system, the frequencies are expected to come as a linear combinations of the fundamental tunes (3 for a 6D system).

To properly study the harmonics of a system, the user should almost always try to unambigously identify the spectral lines, since very close lines can be mistaken for one another and ordering them by amplitude will definitely lead to the wrong results.

Such a categorization of the spectral lines can be done for stable motion from a hamiltonian system like the LHC or any standard map by using the linear combination of fundamental frequencies as a unique ID to follow a given spectral line. See for example the examples/nb_convergence.ipynb notebook for such an approach and the examples/ex_04_regularity_4D.py for an example of the problems which can arise when ordering spectral lines by amplitude.

# Let's assume the following dictionnary of phase space data:
data = {"x": x, "px": px, "y": y, "py": py, "zeta": zeta, "pzeta": pzeta}

# Let's extract the fundamental frequencies
Q_vec = [
    nafflib.tune(data[f"{plane}"], data[f"p{plane}"]) for plane in ["x", "y", "zeta"]
]

# Let's extract some harmonics
A, Q = nafflib.harmonics(x, px, num_harmonics=5)

# Let's find the linear combination of fundamental tunes (Q_vec)
# -----------------
# Note: max_harmonics_order might need to be set to a higher value to find
#       the proper linear combination of frequencies
# -----------------
categorization = nafflib.find_linear_combinations(
    Q, fundamental_tunes=Q_vec, max_harmonic_order=10
)
# -> where categorization = (r_vec,err,combined_frequency)