Bifurcation Module

The bifurcation module provides methods for detecting and analyzing bifurcations in the circular restricted three-body problem.

Base Module

The base module provides the core bifurcation analysis framework.

This module is reserved for future use and will contain the core bifurcation analysis framework.

Analysis Module

The analysis module provides bifurcation analysis methods.

This module is reserved for future use and will contain bifurcation analysis methods.

Transformation Functions

The transforms module provides polynomial transformations for Birkhoff normal form analysis.

_nf2aa_ee()

The _nf2aa_ee() function converts Birkhoff normal form polynomial to action-angle form for elliptic-elliptic libration points (L4, L5).

hiten.algorithms.bifurcation.transforms._nf2aa_ee()[source]

Convert Birkhoff normal form polynomial to action-angle form.

Transforms canonical (q,p) polynomial representation to action-angle (I,theta) representation for elliptic-elliptic libration points (L4, L5). Only monomials with even total canonical degree contribute to integer action exponents.

Parameters:: poly_nf_complex (np.ndarray) – Coefficient array of Birkhoff normal form complex polynomial. Shape corresponds to canonical polynomial degree structure.
Returns:: Action-angle polynomial coefficients. Returns zero array for odd-degree input polynomials (no valid action-angle monomials).
Return type:: np.ndarray
Raises:: ValueError – If polynomial degree cannot be inferred from coefficient array size.

Notes

The transformation uses the canonical-to-action-angle mapping: - Action variables: I_j = (q_j^2 + p_j^2)/2 - Angle variables: theta_j = arctan(p_j/q_j) - Fourier indices: k_j = q_j_exp - p_j_exp - Prefactor: (-i)^(sum of p exponents)

Only processes monomials where each (q_j + p_j) sum is even, ensuring integer action exponents.

_nf2aa_sc()

The _nf2aa_sc() function converts Birkhoff normal form polynomial to action-angle form for saddle-center libration points (L1, L2, L3).

hiten.algorithms.bifurcation.transforms._nf2aa_sc()[source]

Convert Birkhoff normal form polynomial to action-angle form.

Transforms canonical (q,p) polynomial representation to action-angle (I,theta) representation for saddle-center libration points (L1, L2, L3).

The center directions follow identical mapping rules as elliptic-elliptic systems. The hyperbolic direction does not introduce additional angle dependence, so the transformation logic remains unchanged.

Parameters:: poly_nf_complex (np.ndarray) – Coefficient array of Birkhoff normal form complex polynomial.
Returns:: Action-angle polynomial coefficients.
Return type:: np.ndarray

Notes

Currently delegates to _nf2aa_ee() since the transformation rules for center directions are identical. Future extensions could implement saddle-center specific processing (e.g., filtering hyperbolic Fourier harmonics) without affecting the API.