Covariant Lyapunov Vectors

• Covariant Lyapunov vectors are a norm-independent basis that defines intrinsic directions of exponential growth and decay in dynamical systems.
• Algorithms using forward integration with QR and backward iterations extract these vectors to accurately capture the system's invariant subspaces.
• Their computation enhances predictability analysis, hyperbolicity diagnostics, and the identification of coherent structures in high-dimensional dynamics.

Covariant Lyapunov vectors (CLVs) are a norm-independent and dynamically intrinsic basis of the tangent space associated with a trajectory of a dynamical system, providing the physically relevant directions of local instability and stability in phase space. For a given dynamical system, CLVs furnish an “Oseledec splitting” of the tangent bundle into invariant subspaces corresponding to the Lyapunov exponents, and thus offer a rigorous and tangible characterization of the directions in which infinitesimal perturbations grow or decay exponentially. They are essential for quantifying chaos, understanding stability properties, diagnosing hyperbolicity or its violations, interpreting the geometry of invariant manifolds, and supporting advanced analyses in high- and infinite-dimensional systems.

1. Mathematical Foundations and Oseledec Splitting

Given a differentiable dynamical system with state evolution $\Gamma(t)$ governed by $\dot\Gamma = F(\Gamma)$, the linearized evolution of an infinitesimal perturbation $\delta\Gamma$ is given by

$$\frac{d}{dt} \delta\Gamma = J(\Gamma) \delta\Gamma,$$

where $J = \partial F / \partial \Gamma$ is the Jacobian. The tangent space at each point $\Gamma$ is subject to the propagator $D\phi^t$, yielding after time $t$,

$$\delta\Gamma(t) = D\phi^t|_{\Gamma(0)}\,\delta\Gamma(0).$$

The Oseledec multiplicative ergodic theorem guarantees the existence, almost everywhere, of Lyapunov exponents $\lambda_1 > \lambda_2 > ... > \lambda_m$ with corresponding Oseledec subspaces $E^i(\Gamma)$ such that any $v\in E^i(\Gamma)$ obeys

$$\lim_{t \to \pm\infty} \frac{1}{|t|} \ln \|D\phi^t v\| = \pm \lambda_i.$$

The direct sum of these subspaces at almost every point defines the Oseledec splitting: $$T_\Gamma M = E^1(\Gamma) \oplus \cdots \oplus E^m(\Gamma).$$ CLVs are unique (up to nonzero scalar multiples in one-dimensional $E^i$) vectors $v^i(\Gamma)$ spanning these subspaces and satisfying the covariance property: $$D\phi^t|_{\Gamma} v^i(\Gamma) = \alpha^i(t, \Gamma) v^i(\Gamma(t)),$$ with appropriate normalization, and $\alpha^i$ encoding the finite-time growth rate.

2. Algorithms for Computing Covariant Lyapunov Vectors

A numerically tractable algorithm for CLVs, first formalized by Ginelli et al., involves two main phases:

(a) Forward Integration and Orthonormalization

• Integrate the system and the linearized tangent bundle forward in time.
• At each step, apply the Gram–Schmidt (GS) or QR decomposition to a set of $N$ linearly independent vectors, maintaining their orthogonality and populating an upper-triangular matrix $R_n$.
• The orthonormal GS vectors $g^i_n$ converge (with sufficient forward time) to the backward Lyapunov vectors (BLVs) and yield Lyapunov exponents as long-time averages of $\ln \|R_n\|$.

(b) Backward Iteration and Extraction of CLVs

• Store the sequence of $R_n$ (and optionally $g^i_n$) along the trajectory.
• Backward-propagate coefficients $c^{(j,i)}_n$ and reconstruct CLVs as linear combinations of the GS basis: $v_n^{(i)} = \sum_{j=1}^i c_n^{(j,i)} g_n^{(j)}$
• The recursive update for the coefficients uses the inverse of $R_n$ and normalization terms: $C_n = R_{k,n}^{-1} C_{n+k} D_{k,n},$ where $D_{k,n}$ is diagonal with local stretching rates.

Alternative methods (Wolfe–Samelson, LU-based decompositions) find CLVs as orthogonal complements or nullspaces of systems involving both forward and backward Lyapunov vectors but generally require more computations. All methods benefit from the exponential separation governed by Lyapunov exponents: convergence rate is set by $|\lambda_i - \lambda_{i-1}|$.

3. Intrinsic Properties, Symmetry, and Parameterization

CLVs are norm-independent and remain invariant under coordinate transformations up to the appropriate Jacobian action: $v_P^i = \pm \frac{M v_C^i}{\| M v_C^i \|},$ where $M$ is the Jacobian of the transformation. While local (finite-time) exponents in different parameterizations may differ by a total derivative term, the global (asymptotic) exponents are invariant.

In time-reversal-invariant systems (including symplectic/Hamiltonian systems), CLVs preserve pairing symmetries and strict time-reversal symmetry: $$v^{(-),i}(\Gamma) = \pm v^{(+),D+1-i}(\Gamma), \qquad \Lambda^{(-),i}(\Gamma) = -\Lambda^{(+),D+1-i}(\Gamma).$$ The Gram–Schmidt-based exponents and vectors lack this symmetry due to the non-intrinsic orthogonalization process.

4. Localization, Transversality, and Hyperbolicity

CLVs provide a physically meaningful characterization of the geometric structure of the tangent space:

• In high-dimensional or spatially extended systems, CLVs associated with large exponents are strongly spatially localized (particularly in low-density or kinetic regimes), while those corresponding to small exponents (e.g., in Lyapunov “mode” regions) can be delocalized or spatially coherent (Goldstone-like or hydrodynamic modes).
• CLVs themselves are not orthogonal, but are transverse: the angles between CLVs from different Oseledec subspaces are bounded away from zero (except for measure-zero cases such as degenerate spectra).
• The system is hyperbolic if minimum angles between stable and unstable manifolds, as spanned by groups of CLVs, are bounded away from zero. Probability distributions of angles and their scaling with system size provide diagnostic measures of violations or deviations from hyperbolicity (e.g., it is shown that exact tangencies do not occur for generic trajectories).

Mathematically, the tangent space can be partitioned as $TX = E^u \oplus N \oplus E^s$, where $N$ is the neutral (center) manifold.

5. Extensions: Infinite Dimensions, Adjoints, and Data-Driven Computation

CLVs have been rigorously generalized to Hilbert and some Banach spaces using semi-invertible multiplicative ergodic theorems (Noethen, 2019). The convergence rate of the numerical algorithm is governed by the spectral gap between Lyapunov exponents. Tracking and algorithmic implementation in infinite-dimensional settings usually builds on adaptations of the QR/LU-based approaches, with necessary attention to the existence and separation of the Oseledets spaces.

Adjoint covariant Lyapunov vectors (accompanying the primal CLVs) are defined via evolution under the adjoint propagator and are used to compute characteristic, norm-independent angles between vectors, critical for diagnosing tangencies and for use in sensitivity analysis and statistical response theory.

Recent work has enabled the computation of CLVs in purely data-driven settings even when the model equations are unknown. Approaches include identifying local linear (or even nonlinear) models directly from high-dimensional time series using sparse regression (SINDy), then extracting the Jacobian structure and applying standard CLV algorithms (Martin et al., 2021). Data-clustering/VAR-based techniques (e.g., FEM-BV-VAR) have also been used to reconstruct CLVs for multiscale and metastable systems (Viennet et al., 2022). Data assimilation has been exploited to use the filter mean as a surrogate trajectory, permitting computation of approximate CLVs from partial and noisy observations (Roy et al., 2023).

6. Physical Interpretation, Applications, and Implications

The use of CLVs has deepened the interpretation of high-dimensional and complex dynamical systems in several contexts:

• Predictability and Error Growth: CLVs provide the basis for understanding how infinitesimal errors amplify (or contract) along a trajectory; in multiscale or partially hyperbolic systems, they clarify the role of slow, degenerate, or center directions in error propagation and the breakdown of statistical predictability (Vannitsem et al., 2015).
• Hyperbolicity Diagnostics: The ability to compute and analyze angles between invariant manifolds informs whether true hyperbolicity is maintained or violated (e.g., in atmospheric and oceanic flows, molecular dynamics, and plasma).
• Physical Mechanisms: In geophysical fluid dynamics, CLVs have been used to analyze instabilities such as baroclinic/barotropic processes, to quantify the Lorenz energy cycle on physical perturbation subspaces, and to relate local instability to actual energetic exchanges in turbulent flows (Schubert et al., 2014, Schubert et al., 2015).
• Coherent Structures: The geometry of the CLVs, especially the angle field between expanding and contracting directions, has enabled a refined definition of hyperbolic covariant coherent structures (HCCSs) that reflect long-term organizing barriers in fluid flows, contrasting with finite-time Lyapunov exponent diagnostics (Conti et al., 2017).
• Reduced Models and Data Assimilation: In atmospheric and climate models, CLVs (or their unstable subspaces) have been shown to provide effective reduced bases for predictions, ensemble generation, and targeted response analyses.

7. Theoretical and Geometrical Structure

Recent theoretical advances interpret the CLVs as continuous (or differentiable) vector fields (“covariant Lyapunov fields”) over invariant domains, characterized as solutions to a covariant partial differential equation on phase space: $L_F v + b v = 0,$ with a gauge invariance in the normalization of $v(x)$ and corresponding shifts in $b(x)$ directly analogous to gauge transformations in quantum field theory (Marino et al., 2023). The set of CLVs then defines a foliation of phase space, linking dynamical irreversibility, invariant manifolds, and Lyapunov exponents to global geometrical structures.

Table: Key Features and Computational Aspects

Feature	CLVs	Gram–Schmidt/OLVs
Depend on norm	No	Yes
Covariant w/ flow	Yes	No
Orthonormal	No	Yes
Time-reversal symmetric	Yes (if system is)	No
Used for Oseledec splitting	Yes	No
Physical interpretability	Intrinsic/stretching/contraction	Adaptive/rotating

Summary

Covariant Lyapunov vectors provide the primary, norm-independent tool for probing and quantifying the directions of local instability in dynamical systems. Their rigorous mathematical foundation, efficient computational realizability, and robust physical interpretability underlie their central role in modern analyses of chaos, coherent structures, multiscale error dynamics, and advanced methodologies in statistical mechanics, atmospheric and oceanic modeling, and data-driven dynamical systems research.