Covariant Lyapunov vectors

Lyapunov exponents quantify average rates of expansion and contraction, but they do not identify the intrinsic tangent directions associated with those rates. Covariant Lyapunov vectors (CLVs) provide these local directions. Practical algorithms for computing them were introduced independently by Ginelli et al. (2007) and Wolfe and Samelson (2007).

For a continuous flow with tangent propagator \(\mathbf{M}(t_2,t_1)\) from \(t_1\) to \(t_2\), the \(i\)-th CLV satisfies the covariance relation

\[\mathbf{M}(t_2,t_1)\mathbf{v}_i(t_1)=\alpha_i(t_2,t_1)\mathbf{v}_i(t_2),\]

where \(\alpha_i\) is the local expansion or contraction factor. The vectors are ordered by their associated Lyapunov exponents, from \(\lambda_1\) down to the smallest. Unlike the orthonormal vectors generated during the QR calculation of the Lyapunov spectrum, CLVs are generally not orthogonal and transform covariantly with the tangent dynamics. This makes them suitable for studying the geometry of stable, unstable, and center subspaces.

The CLV method implements a Ginelli-style forward and backward algorithm. The tangent dynamics require the Hessians of the Hamiltonian, which the built-in Hénon-Heiles model provides. Because the flow is symplectic, the exponents occur in pairs \(\pm\lambda\) and the vectors associated with \(\lambda_i\) and \(-\lambda_i\) are dynamically related.

Computing the CLVs of the Hénon-Heiles system

Follow a chaotic orbit at energy \(E=1/8\), taking \(x=0\), \(y=-0.2\), \(p_y=0\), and \(p_x>0\) from the energy:

import numpy as np
import matplotlib.pyplot as plt
from pynamicalsys import HamiltonianSystem, PlotStyler

system = HamiltonianSystem(model="henon heiles")
system.integrator("svy4", time_step=0.01)

energy = 1 / 8
x, y, py = 0.0, -0.2, 0.0
potential = (x**2 + y**2) / 2 + x**2 * y - y**3 / 3
px = np.sqrt(2 * (energy - potential) - py**2)

q = [x, y]
p = [px, py]

clvs, clv_trajectory = system.CLV(
    q,
    p,
    2_000.0,
    num_clvs=4,
    tail_time=100.0,
    qr_time_step=0.01,
)

The array clvs has shape (num_samples, 2d, num_clvs), where \(2d=4\) for the two-degree-of-freedom Hénon-Heiles system. At each stored sample, clvs[n, :, i] is the \(i\)-th covariant Lyapunov vector, ordered from the direction of the largest Lyapunov exponent to the smallest, and the matching row of clv_trajectory is [time, x, y, px, py]. Set num_clvs when only the leading vectors are needed.

The time arguments serve different purposes. warmup_time lets the forward orthonormal basis converge before storage begins, and tail_time advances beyond the stored interval to improve the initialization of the backward recursion. The reliability of the vectors near the ends of the stored interval should be checked by increasing warmup_time and tail_time. The seed argument controls the random upper-triangular matrix used to initialize the backward stage.

Angles between covariant directions

The CLV_angles method computes minimum principal angles between subspaces spanned by selected CLVs, requested through subspaces, and direct angles between individual vectors, requested through pairs. For subspaces with orthonormal bases \(\mathbf{Q}_A\) and \(\mathbf{Q}_B\), the minimum principal angle is

\[\theta_{A,B}(t)=\arccos\left[\sigma_{\max}\left(\mathbf{Q}_A^T\mathbf{Q}_B\right)\right].\]

With the default use_abs=True, vector orientation is ignored and the angles lie between \(0\) and \(\pi/2\).

In a Hamiltonian system it is natural to record these angles on a Poincaré section rather than at fixed time steps, which the method does with poincare_section=True. For the chaotic orbit above, sample on the crossings of the \(x=0\) section and request the angles, using zero-based indices,

\[\begin{split}\theta_1 &= \angle\left(\operatorname{span}(\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3),\operatorname{span}(\mathbf{v}_4)\right),\\ \theta_2 &= \angle\left(\operatorname{span}(\mathbf{v}_1),\operatorname{span}(\mathbf{v}_2,\mathbf{v}_3,\mathbf{v}_4)\right),\\ \theta_3 &= \angle\left(\operatorname{span}(\mathbf{v}_1,\mathbf{v}_2),\operatorname{span}(\mathbf{v}_3,\mathbf{v}_4)\right),\\ \theta_4 &= \angle\left(\operatorname{span}(\mathbf{v}_1),\operatorname{span}(\mathbf{v}_4)\right),\end{split}\]

the first three as subspace angles and the last as a direct pair:

total_time = 1_000_000.0

angles, section = system.CLV_angles(
    q,
    p,
    total_time,
    subspaces=[([0, 1, 2], [3]), ([0], [1, 2, 3]), ([0, 1], [2, 3])],
    pairs=(0, 3),
    warmup_time=1e4,
    tail_time=1e4,
    poincare_section=True,
    section_index=0,
    section_value=0.0,
    crossing=1,
)

The columns of angles follow the arguments: the three subspace angles \(\theta_1\), \(\theta_2\), \(\theta_3\), then the pair angle \(\theta_4\). Each row of section is a section point [time, x, y, px, py]. Color the section in the \((y, p_y)\) plane by each angle:

ps = PlotStyler()
ps.apply_style()
fig, ax = plt.subplots(1, 4, sharex=True, sharey=True, figsize=(16, 4))
for index in range(4):
    points = ax[index].scatter(
        section[:, 2],
        section[:, 4],
        c=angles[:, index],
        cmap="plasma",
        vmin=0.0,
        vmax=np.pi / 2,
        s=1,
        edgecolor="none",
    )
    ax[index].set_xlabel("$y$")
ax[0].set_ylabel("$p_y$")
colorbar = fig.colorbar(points, ax=ax, ticks=[0.0, np.pi / 4, np.pi / 2])
colorbar.set_label(r"$\theta$")
colorbar.set_ticklabels(["$0$", r"$\pi/4$", r"$\pi/2$"])
plt.show()
_images/henon_heiles_clv_angles.png

Crossings of the \(x=0\) section of a chaotic Hénon-Heiles orbit at \(E=1/8\), colored by the covariant-vector angles \(\theta_1\), \(\theta_2\), \(\theta_3\), and \(\theta_4\) from left to right.

Angles approaching zero indicate near-tangencies between the selected directions or subspaces, while values bounded away from zero support a transversal splitting along the sampled trajectory. The chaotic orbit visits only the connected chaotic region of the section, leaving the regular islands empty. A finite trajectory cannot by itself prove uniform hyperbolicity, so the minimum angles should be examined over longer intervals and against changes in warmup_time and tail_time. The use of CLV angles as a hyperbolicity diagnostic is discussed by Ginelli et al. (2007) and developed in detail by Kuptsov and Parlitz (2012).

References