Möbiusサドルサイクル：周期倍化と負の固有値#
この notebook は、topology の観点から興味深い fixed point、Möbius saddle cycle を示します。ここでは monodromy matrix DPm が 負の 固有値（\(\lambda_u = -e^{+1/3}\)、\(\lambda_s = -e^{-1/3}\)）を持ちます。
物理的意味：
1 つの完全なトロイダル period（\(m=1\) turn）の後、小さな displacement は stretch/contract しながら 符号を反転 します。元の向きに戻るのは 2 turn 後だけであり、局所 tangent bundle の中に Möbius strip に似た topology が現れます。
これは通常の X-point（正の \(\lambda\)）とは異なり、inverse-hyperbolic または period-doubled fixed point と呼ばれることがあります。
学ぶこと：
Poincaré map に負の固有値を持つ saddle point がどのように現れるか
evolve_DPm_along_cycleで固有値の符号を確認する方法stable/unstable manifold geometry を 3-D で可視化する方法。これらは torus の周りに half-twisted band（Möbius strip）を形成します。
1. Field construction：\((R₀, Z₀)\) にある m=1 Möbius fixed point#
最も単純な場合を扱います。one-turn map の単一 fixed point が \((R_0, Z_0)\) にあり、固有値は \(\lambda_u = -e^{+1/3}\)、\(\lambda_s = -e^{-1/3}\) です。
固有方向は各 turn で \(\pi\) 回転します（dθ/dφ = 1/2）。そのため 1 turn 後に \(\theta \to \theta + \pi\) となり、固有ベクトルが反転します。これが Möbius 的な性質です。
[1]:
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
# ── Fixed point and field parameters ────────────────────────────────────────
R0, Z0 = 1.0, 0.0
BPhi0 = 2.5
m = 1 # one-turn fixed point
# Negative eigenvalues → Möbius / inverse-hyperbolic character
lam_u = -np.e ** (1/3)
lam_s = -np.e ** (-1/3)
print(f"λ_u = {lam_u:.6f}")
print(f"λ_s = {lam_s:.6f}")
print(f"λ_u · λ_s = {lam_u * lam_s:.10f} (should be +1 for area-preserving map)")
print(f"After 2 turns: λ_u² = {lam_u**2:.6f} (now positive, normal hyperbolic)")
# Eigendirections rotate by π per turn (half-integer winding)
dtheta_dphi = 0.5
def theta_u(phi):
return np.pi/2 + phi/2
def theta_s(phi):
return phi/2
# Note: theta_u(2π) = π/2 + π = 3π/2 ≠ theta_u(0)
# The eigenvectors have flipped! This is the Möbius twist.
print(f"\nθ_u(0) = {theta_u(0):.4f} rad = {np.degrees(theta_u(0)):.1f}°")
print(f"θ_u(2π) = {theta_u(2*np.pi):.4f} rad = {np.degrees(theta_u(2*np.pi)):.1f}°")
print(f"Difference = {np.degrees(theta_u(2*np.pi) - theta_u(0)):.1f}° (= 180° flip → Möbius)")
λ_u = -1.395612
λ_s = -0.716531
λ_u · λ_s = 1.0000000000 (should be +1 for area-preserving map)
After 2 turns: λ_u² = 1.947734 (now positive, normal hyperbolic)
θ_u(0) = 1.5708 rad = 90.0°
θ_u(2π) = 4.7124 rad = 270.0°
Difference = 180.0° (= 180° flip → Möbius)
[2]:
# ── Toroidal field model ─────────────────────────────────────────────────────
def BPhi(phi, RZ):
"""Toroidal field: R·B_φ = const."""
return BPhi0 * R0 / RZ[0]
# ── A matrix at the fixed point ──────────────────────────────────────────────
def _eigvec_matrix(phi):
return np.array([
[np.cos(theta_u(phi)), np.cos(theta_s(phi))],
[np.sin(theta_u(phi)), np.sin(theta_s(phi))],
])
def _A_matrix(phi):
"""Jacobian A(φ) of field direction g w.r.t. (R,Z) at the fixed point."""
V = _eigvec_matrix(phi)
# log(|λ|) / (2mπ) gives the per-radian growth rate
mu_u = np.log(np.abs(lam_u)) / (2 * m * np.pi)
mu_s = np.log(np.abs(lam_s)) / (2 * m * np.pi)
Lam = np.diag([mu_u, mu_s])
dV = np.array([
[-dtheta_dphi * np.sin(theta_u(phi)), -dtheta_dphi * np.sin(theta_s(phi))],
[ dtheta_dphi * np.cos(theta_u(phi)), dtheta_dphi * np.cos(theta_s(phi))],
])
return V @ Lam @ np.linalg.inv(V) + dV @ np.linalg.inv(V)
# ── φ-parameterised field direction g = R·B_pol/B_φ ─────────────────────────
def field_g_pol(phi, X_pol):
"""Field direction in (R,Z): g = A(φ) · (X - X_c) + 0 (fixed point has dX_c/dφ=0)."""
Xc = np.array([R0, Z0])
A = _A_matrix(phi)
return A @ (X_pol - Xc)
# Verify: g on the fixed point is zero (it's a FIXED point, not just periodic orbit)
g0 = field_g_pol(0.5, np.array([R0, Z0]))
print("g at fixed point (R0,Z0):", g0, " (should be 0)")
# ── pyna-compatible field function ───────────────────────────────────────────
def pyna_field(rzphi):
R, Z, phi = rzphi[0], rzphi[1], rzphi[2]
X_pol = np.array([R, Z])
g = field_g_pol(phi, X_pol)
# Regularise near the fixed point to avoid division by zero
gnorm2 = np.dot(g, g)
dphi_dl = 1.0 / np.sqrt(R**2 + max(gnorm2, 1e-30))
return np.array([g[0] * dphi_dl, g[1] * dphi_dl, dphi_dl])
print("Field at a nearby point:")
print(pyna_field(np.array([R0 + 0.05, Z0, 0.0])))
g at fixed point (R0,Z0): [0. 0.] (should be 0)
Field at a nearby point:
[-0.00252555 0.0238027 0.95210808]
2. evolve_DPm_along_cycle による DPm の計算#
[3]:
from pyna.topo.monodromy import evolve_DPm_along_cycle
from pyna.topo.toroidal_cycle import ToroidalPeriodicOrbitTrace as PeriodicOrbit
orbit = PeriodicOrbit(
rzphi0=np.array([R0, Z0, 0.0]),
period_m=m,
trajectory=np.zeros((1, 3)),
DPm=np.eye(2),
)
mono = evolve_DPm_along_cycle(
field_func=pyna_field,
orbit=orbit,
dt_output=2 * np.pi / 500,
rtol=1e-11, atol=1e-13,
)
print("DPm (monodromy matrix):")
print(mono.DPm)
print()
print("Eigenvalues:", mono.eigenvalues)
print("Expected: ", np.array([lam_u, lam_s]))
print()
print("Both eigenvalues are NEGATIVE — this is the Möbius saddle character.")
print("Stability index Tr(DPm)/2 =", mono.stability_index,
f" (|k| = {abs(mono.stability_index):.4f} > 1 → hyperbolic)")
DPm (monodromy matrix):
[[-7.16531311e-01 -1.74438498e-09]
[ 1.74438440e-09 -1.39561242e+00]]
Eigenvalues: [-0.71653131 -1.39561242]
Expected: [-1.39561243 -0.71653131]
Both eigenvalues are NEGATIVE — this is the Möbius saddle character.
Stability index Tr(DPm)/2 = -1.0560718675230398 (|k| = 1.0561 > 1 → hyperbolic)
[4]:
# ── Analytic check ───────────────────────────────────────────────────────────
# DPm_analytic = V(2π) · diag(λ_u, λ_s) · V^{-1}(0)
V_end = _eigvec_matrix(2 * m * np.pi)
V_start = _eigvec_matrix(0.0)
DPm_analytic = V_end @ np.diag([lam_u, lam_s]) @ np.linalg.inv(V_start)
print("DPm analytic:")
print(DPm_analytic)
print()
print("Max error:", np.max(np.abs(mono.DPm - DPm_analytic)))
DPm analytic:
[[ 7.16531311e-01 2.12494955e-16]
[-8.77497776e-17 1.39561243e+00]]
Max error: 2.7912248489659737
3. Manifold geometry に現れる Möbius twist の可視化#
Möbius saddle の stable manifold は 3-D で half-twisted band になります。torus の周りを 1 回巻きますが、向きが反転します。2 turn 後には通常の strip になります。
軌道に沿う各 φ で局所 stable/unstable 方向 e_s(φ)、e_u(φ) を描き、half-turn rotation を確認します。
[5]:
phi_plot = np.linspace(0, 2 * np.pi, 400)
# Eigendirection angles along one turn
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].plot(phi_plot / np.pi, np.degrees(theta_u(phi_plot)) % 360, label='θ_u (unstable)', color='tomato')
axes[0].plot(phi_plot / np.pi, np.degrees(theta_s(phi_plot)) % 360, label='θ_s (stable)', color='steelblue')
axes[0].set_xlabel('φ / π')
axes[0].set_ylabel('Eigenvector angle (degrees)')
axes[0].set_title('Eigendirection rotation: half-turn per orbit')
axes[0].legend()
axes[0].axvline(2, color='gray', linestyle='--', label='End of period')
# DX_pol components showing sign flip at φ=2π
axes[1].plot(mono.phi_arr / np.pi, mono.DX_pol_arr[:, 0, 0], label='DX_pol[0,0]', color='steelblue')
axes[1].plot(mono.phi_arr / np.pi, mono.DX_pol_arr[:, 1, 1], label='DX_pol[1,1]', color='tomato')
axes[1].axhline(0, color='k', lw=0.5)
axes[1].axvline(2, color='gray', linestyle='--')
axes[1].set_xlabel('φ / π')
axes[1].set_title('DX_pol diagonal: note values at φ=2π (the DPm)')
axes[1].legend()
plt.tight_layout()
plt.savefig("mobius_DX_pol.png", dpi=120)
plt.show()
[6]:
# ── 3D plot: the stable manifold band (Möbius strip) ─────────────────────────
from mpl_toolkits.mplot3d import Axes3D
nS = 30 # strip width samples
nPhi = 400
Phi = np.linspace(0, 2 * np.pi, nPhi, endpoint=True)
s_vals = np.linspace(-0.25, 0.25, nS) # displacement along stable direction
# Stable manifold: X_c + s · e_s(φ)
theta_s_arr = theta_s(Phi) # angle of stable eigendirection at each φ
e_s_R = np.cos(theta_s_arr) # (nPhi,)
e_s_Z = np.sin(theta_s_arr)
# Grid: [iS, iPhi]
R_strip = R0 + s_vals[:, None] * e_s_R[None, :]
Z_strip = Z0 + s_vals[:, None] * e_s_Z[None, :]
x_strip = R_strip * np.cos(Phi)[None, :]
y_strip = R_strip * np.sin(Phi)[None, :]
fig = plt.figure(figsize=(10, 7))
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x_strip, y_strip, Z_strip,
alpha=0.5, color='steelblue', linewidth=0)
# The fixed point trajectory (a circle in the φ=0 plane → a point)
ax.scatter([R0], [0], [Z0], color='red', s=80, zorder=5, label='Fixed point')
# Unstable manifold band
theta_u_arr = theta_u(Phi)
e_u_R = np.cos(theta_u_arr)
e_u_Z = np.sin(theta_u_arr)
R_strip_u = R0 + s_vals[:, None] * e_u_R[None, :]
Z_strip_u = Z0 + s_vals[:, None] * e_u_Z[None, :]
x_strip_u = R_strip_u * np.cos(Phi)[None, :]
y_strip_u = R_strip_u * np.sin(Phi)[None, :]
ax.plot_surface(x_strip_u, y_strip_u, Z_strip_u,
alpha=0.3, color='tomato', linewidth=0)
ax.set_xlabel('x [m]'); ax.set_ylabel('y [m]'); ax.set_zlabel('Z [m]')
ax.set_title('Möbius saddle: stable (blue) and unstable (red) manifold bands')
ax.legend()
plt.tight_layout()
plt.savefig("mobius_manifold_3d.png", dpi=120)
plt.show()
print()
print("Notice: each strip makes a HALF-TWIST over 2π.")
print("The band only closes (returns to same orientation) after TWO turns.")
Notice: each strip makes a HALF-TWIST over 2π.
The band only closes (returns to same orientation) after TWO turns.
4. Two-turn monodromy：通常の hyperbolic へ戻る#
2 turn（m=2）について DPm を計算すると、負号が打ち消し合い、正の固有値を持つ通常の hyperbolic fixed point が得られます。
[7]:
orbit_2turn = PeriodicOrbit(
rzphi0=np.array([R0, Z0, 0.0]),
period_m=2, # two turns
trajectory=np.zeros((1, 3)),
DPm=np.eye(2),
)
mono_2 = evolve_DPm_along_cycle(
field_func=pyna_field,
orbit=orbit_2turn,
dt_output=2 * np.pi / 500,
rtol=1e-11, atol=1e-13,
)
print("1-turn DPm eigenvalues:", mono.eigenvalues, " (negative)")
print("2-turn DPm eigenvalues:", mono_2.eigenvalues, " (positive, = λ²)")
print()
print("Expected 2-turn eigenvalues:", np.array([lam_u**2, lam_s**2]))
print("Match:", np.allclose(sorted(np.abs(mono_2.eigenvalues)),
sorted([lam_u**2, lam_s**2]), rtol=1e-4))
1-turn DPm eigenvalues: [-0.71653131 -1.39561242] (negative)
2-turn DPm eigenvalues: [0.51341713 1.94773399] (positive, = λ²)
Expected 2-turn eigenvalues: [1.94773404 0.51341712]
Match: True
5. まとめ#
Property |
Value |
|---|---|
Fixed point |
\((R_0, Z_0) = (1.0, 0.0)\) |
Orbit period |
\(m = 1\) turn |
DPm eigenvalues |
\(\lambda_u = -e^{+1/3}\), \(\lambda_s = -e^{-1/3}\) |
Character |
Möbius saddle（inverse-hyperbolic） |
Eigenvector rotation |
\(\Delta\theta = \pi\) per turn（half-twist） |
Two-turn map |
\(\lambda^2 = e^{\pm 2/3}\) を持つ normal hyperbolic |
重要な insight： DPm 固有値の符号は、不変多様体の topological type を encode します。単一文字 M ではこの点が隠れます。DPm は、これが full-period Poincaré map derivative であり、その spectrum の性質が局所 topology を決めることを明示します。
次のステップ：
elliptic orbit case については
monodromy_xcycle_analytic.ipynbを参照してください実機応用例については
pyna/notebooks/research/を参照してください