Sensitivity and the Lorenz Attractor

Copyright (C) 2026 Andreas Kloeckner

MIT License

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from scipy.integrate import solve_ivp

def log_norm_2(A):
    """
    Logarithmic norm (matrix measure) μ₂(A) = λ_max((A + Aᵀ)/2).
    This is the tightest Lipschitz constant available from the 2-norm:
    ‖e^{At}‖₂ ≤ exp(μ₂(A)·t).
    """
    S = 0.5 * (A + A.T)
    return np.max(np.linalg.eigvalsh(S))
 

SIGMA = 10.0
RHO   = 28.0
BETA  = 8.0 / 3.0
 
def lorenz(t, y):
    x, y_, z = y
    return np.array([
        SIGMA * (y_ - x),
        x * (RHO - z) - y_,
        x * y_ - BETA * z,
    ])
 
def lorenz_jac(y):
    """Jacobian of the Lorenz vector field at y."""
    x, y_, z = y
    return np.array([
        [-SIGMA,  SIGMA,  0.0  ],
        [RHO - z, -1.0, -x    ],
        [y_,       x,  -BETA  ],
    ])
 

T_END      = 20
# T_END      = 80
EPSILON    = 1e-6
# EPSILON    = 1e-2

Y0         = np.array([1.0, 0.0, 0.0])
Y0_PERT    = Y0 + EPSILON * np.array([1.0, 0.0, 0.0])
 
t_eval = np.linspace(0, T_END, 8000)
 
print("Integrating reference trajectory …")
sol_ref  = solve_ivp(lorenz, [0, T_END], Y0,      t_eval=t_eval,
                     method='RK45', rtol=1e-10, atol=1e-12)
print("Integrating perturbed trajectory …")
sol_pert = solve_ivp(lorenz, [0, T_END], Y0_PERT, t_eval=t_eval,
                     method='RK45', rtol=1e-10, atol=1e-12)
 
t = sol_ref.t
Y = sol_ref.y      # shape (3, N)
Yp = sol_pert.y

Integrating reference trajectory …
Integrating perturbed trajectory …

fig = plt.figure(figsize=(14,10))
ax3d = plt.gcf().add_subplot(projection='3d')
 
ax3d.plot(*Y,  lw=0.5, alpha=0.7, label="reference")
ax3d.plot(*Yp, lw=0.5, alpha=0.7, label=f"perturbed (ε={EPSILON:.0e})")
ax3d.scatter(*Y[:, 0],   s=20, zorder=5)
ax3d.scatter(*Yp[:, 0],  s=20, zorder=5)
 
ax3d.set_xlabel("x", )
ax3d.set_ylabel("y", )
ax3d.set_zlabel("z", )
ax3d.legend(fontsize=7, loc="best")
 

<matplotlib.legend.Legend at 0x7f662908b0e0>

separation = np.linalg.norm(Y - Yp, axis=0)
plt.plot(t, separation)

[<matplotlib.lines.Line2D at 0x7f6628db6660>]

# (a) Tight bound: integrate μ₂(J(y(t))) along the reference orbit.
#     ‖δy(t)‖ ≤ ε · exp(∫₀ᵗ μ₂(J(y(s))) ds)
print("Computing logarithmic norms along orbit …")
log_norms = np.array([log_norm_2(lorenz_jac(Y[:, i])) for i in range(len(t))])
# Cumulative integral via trapezoidal rule
log_norm_integral = np.zeros_like(t)
log_norm_integral[1:] = np.cumsum(
    0.5 * (log_norms[:-1] + log_norms[1:]) * np.diff(t)
)
bound_tight = EPSILON * np.exp(log_norm_integral)
 
# (b) Crude bound: use the supremum of μ₂ over the observed orbit.
L_crude = np.max(log_norms)
bound_crude = EPSILON * np.exp(L_crude * t)
 
print(f"  max log-norm along orbit: {L_crude:.4f}")
print(f"  (compare: largest Lyapunov exponent ≈ 0.906)")
 

Computing logarithmic norms along orbit …
  max log-norm along orbit: 14.0256
  (compare: largest Lyapunov exponent ≈ 0.906)

 
plt.semilogy(t, separation,     lw=1.5, label="actual ‖δy(t)‖")
plt.semilogy(t, bound_tight,     lw=1.2, ls="--",
             label=r"tight P-L: $\varepsilon\exp\!\left(\int_0^t \mu_2(J)\,ds\right)$")
plt.semilogy(t, bound_crude,      lw=1.0, ls=":",
             label=rf"crude P-L: $\varepsilon\exp(L_\max t)$, $L={L_crude:.2f}$")
 
plt.xlabel("t")
plt.ylabel("‖y(t) − ŷ(t)‖")
plt.legend(loc="best")

<matplotlib.legend.Legend at 0x7f6628b37cb0>