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Theory and computation of covariant Lyapunov vectors

Pavel V. Kuptsov, Ulrich Parlitz

arXiv:1105.5228v3nlin.CD

TL;DR

The paper addresses how to characterize perturbation directions in complex high-dimensional dynamics when conventional Lyapunov-vector concepts are limited. It synthesizes their theory and computation, introduces adjoint covariant vectors, and develops an improved computational approach. The resulting framework relates covariant-vector geometry to norm-independent characteristic numbers and supports hyperbolicity tests without explicitly computing covariant vectors.

  • Problem

    High-dimensional nonlinear dynamics is difficult to analyze, while forward and backward Lyapunov vectors are non-covariant and norm-dependent.

  • Method

    The paper systematically explains Lyapunov-vector theory and algorithms, introduces adjoint covariant vectors, and improves covariant-vector computation using an optimized LU decomposition of a scalar-product matrix.

  • Results

    The analysis establishes adjoint covariant vectors whose angles with covariant vectors are norm-independent characteristic numbers and presents a more efficient covariant-vector method.

  • Takeaways & Limitations

    Angles between corresponding covariant and adjoint covariant vectors can compactly represent covariant-vector information and indicate homoclinic tangencies through orthogonality.

  • Takeaways & Limitations

    Forward and backward Lyapunov vectors depend on the chosen tangent-space norm and are not covariant with the dynamics.

Abstract

from arXiv · show

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.

INTRODUCTION

The introduction frames Lyapunov analysis as a way to study complex high-dimensional dynamics, then distinguishes several perturbation-direction concepts and motivates covariant Lyapunov vectors as norm-independent, dynamically covariant directions.

  • Lyapunov exponents quantify perturbation growth in state-space directions and relate to properties including sensitivity to initial conditions, entropy production, and attractor dimension.
  • Tangent-space dynamics is generated by linear propagators, making its analysis applicable across broad classes of nonlinear systems.
  • Bred, singular, optimal, finite-time normal, forward, and backward vectors provide alternative descriptions of perturbation directions, but forward and backward vectors are non-covariant and norm-dependent.
  • Covariant Lyapunov vectors are nonorthogonal, norm-independent, invariant under time reversal, and covariant with the dynamics, while numerical approximations can lose perfect covariance through accumulated errors.
  • The paper develops theoretical foundations and numerical algorithms for Lyapunov vectors, including an efficient covariant-vector method and adjoint covariant vectors whose angles provide norm-independent characteristic numbers.

B. Properties of propagators. Transformation of volumes built on singular vectors

The section explains how tangent propagators transform finite- and infinite-time volumes through singular values, and how these limits define Lyapunov exponents and forward or backward Lyapunov vectors.

  • Finite-time volume transformation: A forward propagator maps right singular vectors at t1 to left singular vectors at t2, while singular values determine the stretching of each direction.
  • Finite-time volume transformation: A k-dimensional volume evolves by the product of its first k singular values, whose logarithmic stretch ratios act as local Lyapunov exponents.
  • Finite-time volume transformation: Backward evolution via F(t1,t2)^-1 reverses the construction: left singular vectors at t2 map to right singular vectors at t1 with reciprocal stretching factors.
  • Infinite-time limits: As time intervals become infinite, singular vectors and stretch rates converge to Lyapunov vectors and exponents through the Oseledec multiplicative ergodic theorem.
  • Infinite-time limits: Forward and backward Lyapunov vectors are orthonormal and time-dependent, remain norm-dependent, and correspond to Lyapunov exponents with opposite signs in the backward formulation.
  • Volume growth and degeneracy: Average growth rates of volumes built from forward Lyapunov vectors equal sums of the corresponding Lyapunov exponents, with degeneracy making vector choices nonunique.

D. Oseledec subspaces. Asymptotic behavior of arbitrary vectors and volumes

Oseledec subspaces organize vectors and volumes by asymptotic growth rates: generic forward iterations select leading backward directions, while subspace structure governs higher-dimensional volume growth and embeddings.

  • Oseledec subspaces: Oseledec subspaces consist of vectors whose forward or backward exponential rates are bounded by the associated Lyapunov exponent thresholds.
  • Asymptotic behavior of vectors: Almost any vector asymptotically grows or decays with the leading exponent and approaches the corresponding leading backward Lyapunov direction.
  • Asymptotic behavior of volumes: For a nondegenerate leading exponent, almost any square grows at rate λ1+λ2 and its directions approach the relevant backward Oseledec subspace.
  • Degenerate growth rates: When the leading exponent has multiplicity two, a square grows at rate 2λ(1), while a cube grows at rate 2λ(1)+λ(2)=λ1+λ2+λ3.
  • General volume behavior: In general, almost any k-dimensional volume grows or decays at a rate given by the sum of the first k Lyapunov exponents, in forward or backward time as appropriate.

E. Finite-time evolution of forward and backward Lyapunov vectors

Finite-time propagators transform forward and backward Lyapunov vectors into non-Lyapunov vectors, so triangular factorizations are required to recover the corresponding orthogonal vector sets.

  • Degenerate Lyapunov spectra make the forward-vector matrix non-unique, but the orthogonal matrix in the QL factorization remains uniquely determined.
  • Forward Lyapunov vectors are mapped into vectors that generally are not forward Lyapunov vectors, making them non-covariant under the dynamics.
  • A QL factorization recovers forward Lyapunov vectors by determining the earlier-time orthogonal matrix uniquely from the later-time one.
  • Backward Lyapunov vectors obey an analogous triangular relation, with an upper triangular non-singular matrix, and are likewise non-covariant.
  • Orthogonalization constructs each q_i within the span of the first i mapped vectors, preserving the nested subspaces represented by the original vectors.

II. NUMERICAL COMPUTATION OF LYAPUNOV EXPONENTS AND FORWARD AND BACKWARD VECTORS

Numerical algorithms use iterative propagation and orthogonalization to compute Lyapunov exponents and forward or backward vectors, while algorithm choice determines which spectral components are most accurate.

  • Directly computing the asymptotic propagator is impractical because long-time evolution aligns almost every vector with the first backward Lyapunov vector.
  • Repeated propagation by F and QR factorization makes columns of Q converge to backward Lyapunov vectors, including any desired number of leading vectors.
  • Degenerate exponents leave particular vector directions dependent on the initial matrix, although the resulting vectors remain within the corresponding degenerate subspace.
  • QR factorization separates mapped vectors into an orthogonal matrix and an upper triangular matrix whose diagonal projections determine local Lyapunov exponents.
  • Adjoint-propagator iterations compute backward vectors in reverse order, while backward iterations with G^-1 or F^-1 compute forward vectors.
  • All four algorithms are most precise for dominating exponents and vectors, while combining complementary algorithms improves accuracy across the full spectrum.

III. COVARIANT LYAPUNOV VECTORS

Covariant Lyapunov vectors are norm-independent directions that lie in intersections of Oseledec subspaces and remain covariant under tangent dynamics. Their adjoint counterparts provide norm-independent angle-based characteristics, while vector merging signals invariant-manifold tangencies and non-hyperbolicity.

  • III. COVARIANT LYAPUNOV VECTORS: Forward and backward Lyapunov vectors preserve their associated subspaces but require triangular transformations and are not individually covariant.Covariant vectors instead evolve by diagonal factors that preserve their directions under tangent flow.
  • III. COVARIANT LYAPUNOV VECTORS: Adjoint covariant vectors are covariant under adjoint dynamics, and their corresponding angles with original covariant vectors are norm-independent.The diagonal elements of D are cosines of these corresponding angles and can be treated as characteristic numbers through their statistics.
  • III. COVARIANT LYAPUNOV VECTORS: Covariant Lyapunov vectors lie in intersections of Oseledec subspaces and are norm-independent, unlike forward and backward Lyapunov vectors.They are defined to be covariant with the propagator and invariant under time reversal.
  • III. COVARIANT LYAPUNOV VECTORS: If Lyapunov exponents are degenerate, covariant vectors are not unique because multiple triangular factorizations can produce different vector sets.The ambiguity occurs if and only if Lyapunov exponents are degenerate.
  • III. COVARIANT LYAPUNOV VECTORS: Merging covariant vectors makes them collinear, produces singular triangular and vector matrices, and indicates tangencies associated with non-hyperbolic chaotic attractors.The merging is time-invariant: vectors identical at one time remain identical under the tangent flow.
  • III. COVARIANT LYAPUNOV VECTORS: Each adjoint covariant vector is orthogonal to all noncorresponding covariant vectors, while tangencies make the corresponding diagonal elements of D vanish.Adjoint vectors can be found as null vectors of the matrix formed by all covariant vectors except the target vector.

A. Intersection of Oseledec subspaces

The intersection method constructs covariant vectors from forward and backward Lyapunov vectors by intersecting Oseledec subspaces. Although it applies to the full spectrum, it is inefficient when only a few leading vectors are needed.

  • A. Intersection of Oseledec subspaces: Covariant vectors can be obtained by intersecting Oseledec subspaces constructed from forward and backward Lyapunov vectors.The intersection is computed using principal angles based on singular values and vectors of submatrices of P.
  • A. Intersection of Oseledec subspaces: Computing the jth covariant vector requires the first j backward vectors and the last m −j + 1 forward vectors.These are generated using forward iterations with F and inverse iterations with F−1.
  • A. Intersection of Oseledec subspaces: The method always requires m+1 forward and backward vectors, so it is suitable for the whole spectrum but inefficient for only a few leading covariant vectors.Its accuracy is approximately flattened across the spectrum because backward and forward vectors are most accurate at opposite ends.

B. Method of LU factorization

The LU-based approach computes covariant vectors from null spaces of rectangular submatrices, avoiding assumptions that triangular factors remain nonsingular. It supports partial-spectrum computation and provides diagnostics for vector merging and hyperbolicity.

  • B. Method of LU factorization: The improved approach avoids computing the whole forward or backward Lyapunov spectrum when only a few leading covariant vectors are required.It is presented alongside the earlier Wolfe–Samelson and Ginelli approaches as a new method for this problem.
  • B. Method of LU factorization: Writing P = A+(A−)−1 identifies an LU factorization whose triangular factors encode the covariant vectors, but standard LU routines fail when these factors are singular.Row and column permutations used by standard routines are unsuitable because the ordering in P is essential.
  • B. Method of LU factorization: The jth covariant vector is computed from the null space of P(1 : j −1, 1 : j), then normalized to obtain a unit vector.The method remains valid despite possible singularities of the triangular factors.
  • B. Method of LU factorization: Multiple null-space solutions occur only under Lyapunov-exponent degeneracy, allowing one solution to be chosen arbitrarily.This reflects the non-uniqueness of covariant vectors associated with degenerate exponents.
  • B. Method of LU factorization: Singularity of P(1 : j, 1 : j) is necessary and sufficient for merging of the jth and (j + 1)th covariant vectors.Such merging indicates tangencies of invariant manifolds and can occur on chaotic non-hyperbolic attractors.
  • B. Method of LU factorization: The method can study near-singularity across all submatrices P(1 : j, 1 : j) and compute selected entries of the covariant–adjoint angle matrix D.Only a few leading elements of D are needed when only a few leading vectors are required.
  • B. Method of LU factorization: For trajectory series, implementation stores trajectory points and backward vectors, then integrates backward after a transient convergence stage.QR factorizations, interpolation, and SVD-based null-space calculations are among the required numerical components.
  • B. Method of LU factorization: Figure 4 contrasts LU factorization, the Wolfe–Samelson orthogonal-complement method, and Ginelli’s iterative method for computing covariant vectors.The first two methods appear in panel a, while Ginelli’s method appears in panel b.

C. Orthogonal complement method of Wolfe and Samelson

The Wolfe–Samelson method reduces the data needed for leading covariant vectors by replacing a subspace of last forward vectors with an orthogonal complement. The resulting equations are equivalent to the LU method for the needed solution.

  • C. Orthogonal complement method of Wolfe and Samelson: The Wolfe–Samelson method uses the local covariant-vector relation and orthogonality of P to compute leading covariant vectors efficiently.Its formulation avoids dependence on the last rows of P.
  • C. Orthogonal complement method of Wolfe and Samelson: The needed subspace spanned by the last m−j+1 forward vectors is obtained as the orthogonal complement of the first j−1 vectors.This replaces direct computation of the full last-vector block.
  • C. Orthogonal complement method of Wolfe and Samelson: Pazó et al. modified the Wolfe–Samelson method using QR factorizations and backward iterations with the transposed propagator.These procedures are part of the standard computation of forward and backward Lyapunov vectors.
  • C. Orthogonal complement method of Wolfe and Samelson: The matrix equation P(1 : j −1, 1 : j)ᵀP(1 : j −1, 1 : j)A−(1 : j, j) = 0 has solutions containing those of the LU equation.Because only one solution is needed at each j, the LU method produces the same result while avoiding redundant matrix multiplication.

D. Backward iterations, method of Ginelli et al.

Ginelli et al.’s method computes covariant Lyapunov vectors through backward iterations in the space of projections onto backward Lyapunov vectors, while practical implementations balance convergence, accuracy, speed, and memory.

  • Core idea: Backward iterations with the inverse upper-triangular matrix RF^-1 converge to the covariant-vector coefficients in the backward Lyapunov basis.The method exploits the equivalence between projected tangent-space iterations and iterations with RF^-1.
  • Implementation: Only the first j covariant vectors need be computed when the corresponding backward Oseledec subspaces are sufficient.The same asymptotic structure also supports computation of the first j adjoint covariant vectors through a transposed-propagator procedure.
  • Implementation: The procedure first computes and stores backward Lyapunov vectors and triangular matrices, then performs backward iterations to recover covariant vectors at desired trajectory points.The stored interval and transient length determine where the converged vectors can later be computed.
  • Numerical limitations: Ill-conditioned RF can reduce accuracy when many covariant vectors are computed in strongly contracting systems, especially for vectors associated with large negative exponents.Implicit iterations and shorter QR-orthogonalization intervals are recommended, while errors from small diagonal elements primarily affect minor vectors.
  • Numerical limitations: Near tangencies of invariant manifolds can create numerical zeros that falsely persist as exact tangencies, requiring a small noise perturbation to cure the artifact.The problem arises because small diagonal values in A(tn) accumulate and underflow during iteration.
  • Memory and speed: For m = 100, KAB = 1000, and double precision, the LU method requires approximately 76 megabytes, compared with approximately 118 megabytes for backward iterations at kBC = 1.Backward iterations also have a transient-memory term growing as kBCm^3b/2, making transient minimization important.

V. EXAMPLES

The constant-Jacobian example makes the Lyapunov-vector constructions explicit: exponents coincide with eigenvalue magnitudes, while covariant and adjoint vectors follow from eigenvectors of J and -J^T. The example also verifies that the LU, Wolfe–Samelson, and Ginelli constructions recover the same covariant vectors.

  • Constant-Jacobian system: For a time-independent Jacobian with real eigenvalues, the Lyapunov exponents are λ1,2,3 = 1, −1, −3, and the eigenvectors of J are covariant Lyapunov vectors.Eigenvectors of -J^T provide the corresponding adjoint covariant vectors.
  • Constant-Jacobian system: Forward and backward Lyapunov vectors arise as eigenvectors of far-future and far-past operators, respectively, with norms held constant when taking the limits.This construction connects the limiting operators to the Lyapunov-vector definitions used in the example.
  • Constant-Jacobian system: The first backward and last forward Lyapunov vectors coincide with the first and last covariant vectors, and the limit-operator eigenvalue logarithms reproduce the Lyapunov exponents.The example therefore checks the expected relationships among the different vector constructions.
  • LU method: The LU construction obtains A− from algebraic equations, while B+ for adjoint covariant vectors is constructed analogously using the transposed matrix.Both equation systems include unit-column-norm constraints.
  • Method verification: The computed matrices satisfy Γ = Φ−A− and Θ = Φ+B+, confirming the defining relations for covariant and adjoint covariant vectors.The Wolfe–Samelson construction and Ginelli’s method likewise reproduce the same covariant vectors in this example.

B. Generalized Hénon map

The generalized Hénon map illustrates how covariant and adjoint covariant vectors characterize hyperchaotic dynamics, homoclinic tangencies, and non-hyperbolicity.

  • Generalized Hénon map: For a = 1.76 and b = 0.1, the generalized three-dimensional Hénon map generates a hyperchaotic attractor with three stated Lyapunov exponents.The exponents are λ1 = 0.225, λ2 = 0.188, and λ3 = −2.716.
  • Generalized Hénon map: The attractor visualization colors points by det[P(1 : 2, 1 :2)], with dark red regions indicating almost tangent covariant Lyapunov vectors.These regions correspond to an almost singular submatrix P(1 : j, 1 : j) with j = 2.
  • Generalized Hénon map: Adjoint covariant vectors provide norm-independent angles with original covariant vectors, and orthogonality indicates homoclinic tangencies between stable and unstable manifolds.Such tangencies are characteristic of non-hyperbolic chaos.
  • Generalized Hénon map: The paper presents an efficient covariant-vector method based on an optimized LU decomposition of the scalar-product matrix P.The approach explicitly formulates P and avoids some redundant computations relative to the method by Wolfe and Samelson.
  • Generalized Hénon map: Non-hyperbolicity can be detected without explicitly computing covariant vectors by identifying nearly singular j × j submatrices of P along a trajectory.Here, j is the number of positive Lyapunov exponents.
  • Generalized Hénon map: When Lyapunov exponents are degenerate, all Lyapunov-vector types are non-unique, although standard algorithms remain usable for forward and backward vectors.Different seed matrices can yield different vector sets, with any one considered appropriate.

Appendix: Pseudocode for the LU method

The appendix specifies the LU-method workflow, from trajectory and vector initialization through forward propagation, storage, backward integration, and construction of covariant vectors.

  • Appendix: Pseudocode for the LU method: The pseudocode takes the number of covariant vectors, stored trajectory points, tangent-space dimension, orthogonalization interval, and convergence-step counts as inputs.The convergence counts separately cover attraction, forward-vector, and backward-vector stages.
  • Appendix: Pseudocode for the LU method: Required subroutines include basic-system solving, forward and transposed propagator actions, null-vector computation, QR orthogonalization, transposition, random generation, and matrix multiplication.These operations support the forward and backward stages of the method.
  • Appendix: Pseudocode for the LU method: The algorithm returns an array of stored m by nclv matrices whose columns are the computed covariant Lyapunov vectors.The output is indexed over the nstore trajectory points.
  • Appendix: Pseudocode for the LU method: The preliminary stage initializes a random orthogonal matrix Q and advances the basic system while repeatedly orthogonalizing Q.The trajectory first spends nspend_att steps converging to the attractor.
  • Appendix: Pseudocode for the LU method: Stage A-B propagates Q forward, orthogonalizes it, advances the state, and stores both trajectory points and forward-vector matrices.The stored arrays are PhiMns and traj.
  • Appendix: Pseudocode for the LU method: Stage B-C extends the trajectory backward-stage storage, while Stage C-B recreates Q with one fewer column before iterating backward.The backward loop proceeds from nspend_bkw down to 2.
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