Tutorial: Möbius Saddle Cycle — Period-Doubling and Negative Eigenvalues#

This notebook demonstrates a topologically interesting fixed point: a Möbius saddle cycle, where the monodromy matrix DPm has negative eigenvalues (\(\lambda_u = -e^{+1/3}\), \(\lambda_s = -e^{-1/3}\)).

Physical meaning:
After one full toroidal period (\(m=1\) turn), a small displacement flips sign as it stretches/contracts. It only returns to its original orientation after two turns — a Möbius-strip-like topology in the local tangent bundle.

This is distinct from the ordinary X-point (positive \(\lambda\)) and is sometimes called an inverse-hyperbolic or period-doubled fixed point.

What you will learn:

  1. How negative-eigenvalue saddle points arise in Poincaré maps

  2. Using evolve_DPm_along_cycle to confirm the eigenvalue sign

  3. Visualising the stable/unstable manifold geometry in 3-D — they form a half-twisted band (Möbius strip) around the torus

1. Field construction — m=1 Möbius fixed point at (R₀, Z₀)#

The simplest case: a single fixed point of the one-turn map at \((R_0, Z_0)\) with eigenvalues \(\lambda_u = -e^{+1/3}\), \(\lambda_s = -e^{-1/3}\).

The eigendirections rotate by \(\pi\) each turn (dθ/dφ = 1/2), so after one turn \(\theta \to \theta + \pi\) — the eigenvectors flip, hence the Möbius character.

[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. Computing DPm with evolve_DPm_along_cycle#

[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. Visualising the Möbius twist in the manifold geometry#

The stable manifold of a Möbius saddle is a half-twisted band in 3-D: it wraps around the torus once but flips orientation. After two turns it becomes a regular strip.

We draw the local stable/unstable directions e_s(φ), e_u(φ) at each φ along the orbit to see the half-turn rotation.

[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: back to normal hyperbolic#

If we compute DPm for 2 turns (m=2), the negative signs cancel and we recover a normal hyperbolic fixed point with positive eigenvalues.

[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. Summary#

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

Normal hyperbolic with \(\lambda^2 = e^{\pm 2/3}\)
Key insight: The sign of DPm eigenvalues encodes the topological type of the invariant manifolds. A single letter M obscures this;
DPm makes explicit that it is the full-period Poincaré map derivative, whose spectral properties determine the local topology.

Next steps:

  • See monodromy_xcycle_analytic.ipynb for the elliptic orbit case

  • See pyna/notebooks/research/ for real W7-X and EAST applications