code/floquet.py

"""Exact-formula Floquet optimizer for y'' + u(s)y = 0, 1 <= u <= R**2.

Floating-point evaluation is not a formal proof or an interval certificate.
All times passed to this module are dimensionless.
"""
from __future__ import annotations
from dataclasses import dataclass
import math
from typing import Sequence
import numpy as np
from scipy.optimize import brentq

@dataclass(frozen=True)
class Candidate:
    winding: int
    eta: float
    high_time: float
    low_time: float
    log_gain: float


def cell_times(R: float, eta: float) -> tuple[float, float]:
    """High-u then low-u residence times; numerically stable at large |eta|."""
    if not math.isfinite(R) or R <= 1 or not math.isfinite(eta):
        raise ValueError('Require finite R > 1 and finite eta.')
    if eta >= 0:
        low = math.atan2(1.0, eta)
        high = (math.pi - math.atan2(R, eta)) / R
    else:
        low = math.pi - math.atan2(1.0, -eta)
        high = math.atan2(R, -eta) / R
    return high, low


def cell_gain(R: float, eta: float) -> float:
    if R <= 1:
        raise ValueError('R must exceed one.')
    return 0.5 * math.log1p((R * R - 1.0) / (1.0 + eta * eta))


def solve_eta(R: float, period: float) -> float:
    if not math.pi / R < period < math.pi:
        raise ValueError('Require pi/R < cell period < pi.')
    bracket = 1.0
    while sum(cell_times(R, -bracket)) < period or sum(cell_times(R, bracket)) > period:
        bracket *= 2.0
        if bracket > 1e15:
            raise ArithmeticError('Period is too close to an endpoint for this float implementation.')
    return float(brentq(lambda a: sum(cell_times(R, a)) - period,
                        -bracket, bracket, xtol=2e-13, rtol=2e-14))


def fixed_period(R: float, S: float) -> tuple[float, list[Candidate]]:
    """Return the sharp log-spectral-radius bound and every positive candidate.

    Winner ties must not be broken into a mathematical uniqueness claim by
    floating-point tolerance. Extremely near resonance endpoints use exact
    arithmetic or certified interval evaluation instead.
    """
    if not math.isfinite(R) or R <= 1 or not math.isfinite(S) or S <= 0:
        raise ValueError('Require finite R > 1 and S > 0.')
    candidates = []
    for n in range(1, math.floor(R * S / math.pi) + 1):
        p = S / n
        if not math.pi / R < p < math.pi:
            continue
        a = solve_eta(R, p)
        h, l = cell_times(R, a)
        candidates.append(Candidate(n, a, h, l, n * cell_gain(R, a)))
    return max((c.log_gain for c in candidates), default=0.0), candidates


def free_period(R: float) -> Candidate:
    """Unique root G(eta) = eta*tau(eta); eta is the optimal growth rate."""
    if R <= 1:
        raise ValueError('R must exceed one.')
    a = float(brentq(lambda e: cell_gain(R, e) - e * sum(cell_times(R, e)),
                    0.0, R, xtol=2e-14))
    h, l = cell_times(R, a)
    return Candidate(1, a, h, l, cell_gain(R, a))


def transfer(u: float, duration: float) -> np.ndarray:
    if not math.isfinite(u) or u <= 0 or not math.isfinite(duration) or duration < 0:
        raise ValueError('Require u > 0 and duration >= 0, both finite.')
    w = math.sqrt(u)
    c, s = math.cos(w * duration), math.sin(w * duration)
    return np.array([[c, s / w], [-w * s, c]], dtype=float)


def monodromy(controls: Sequence[float], durations: Sequence[float]) -> np.ndarray:
    if len(controls) != len(durations) or not len(controls):
        raise ValueError('Control and duration lists must have the same positive length.')
    M = np.eye(2)
    for u, t in zip(controls, durations):
        M = transfer(float(u), float(t)) @ M
    return M


def log_spectral_radius(M: np.ndarray) -> float:
    """For a real determinant-one 2x2 monodromy; not stable for huge products."""
    if np.asarray(M).shape != (2, 2):
        raise ValueError('Expected a 2x2 matrix.')
    d = abs(float(np.trace(M))) / 2.0
    return math.acosh(d) if d > 1.0 else 0.0


def phase_speed(theta: float, u: float) -> float:
    return u * math.cos(theta)**2 + math.sin(theta)**2


def radial_speed(theta: float, u: float) -> float:
    return (u - 1.0) * math.sin(theta) * math.cos(theta)


def phase_selector(theta: float, eta: float, R: float) -> float:
    c, s = math.cos(theta), math.sin(theta)
    return R * R if c * s + eta * c * c > 0 else 1.0


def defect_integrand(theta: float, u: float, eta: float, R: float) -> float:
    star = phase_selector(theta, eta, R)
    switch = abs(math.sin(theta) * math.cos(theta) + eta * math.cos(theta)**2)
    return switch * abs(u - star) / (phase_speed(theta, u) * phase_speed(theta, star))