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The Lyapunov Characteristic Exponents and their computation

Charalampos Skokos

arXiv:0811.0882v2nlin.CDastro-phmath-phmath.DSmath.NAphysics.comp-ph

TL;DR

LCEs quantify asymptotic perturbation growth, motivating numerical chaos indicators and spectrum computation. This survey develops their theoretical basis and reviews computational methods for conservative systems, with briefer coverage of dissipative systems and time series.

  • Problem

    The paper addresses how LCE theory and numerical techniques can characterize perturbation growth and distinguish chaotic from regular orbits.

  • Method

    The survey explains the MET and reviews algorithms for computing maximal, few, or all LCEs, including standard, QR, and SVD-based methods.

  • Results

    The survey establishes the MET as the theoretical basis for LCE computation and relates the standard method to QR decomposition.

  • Takeaways & Limitations

    The review connects LCE computation with chaos detection through the maximal LCE and with broader spectrum-based analysis.

Abstract

from arXiv · show

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68}, which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

2 Autonomous Hamiltonian systems and symplectic maps . . . 7

This section introduces autonomous Hamiltonian systems and symplectic maps, followed by historical notes on early LCE computations.

  • The section presents autonomous Hamiltonian systems and symplectic maps as foundational dynamical-system settings.
  • It also reviews early attempts to apply theoretical results to the numerical evaluation of LCEs.

4 Lyapunov Characteristic Exponents: Theoretical treatment 14

This part develops the theoretical treatment of LCEs and then organizes numerical computation of maximal, partial, and complete spectra alongside chaos-detection and broader applications.

  • The theoretical treatment covers definitions, basic theorems, LCEs of different orders, the MET, and properties of the LCE spectrum.
  • A dedicated section examines computation of the maximal LCE and its behavior for regular and chaotic orbits.
  • The spectrum-computation section presents the standard method, QR connections, other methods, and computation of the complete spectrum.
  • Further sections address chaos-detection techniques and LCE computation for dissipative systems and time series.

1 Introduction

The introduction motivates LCEs as measures of perturbation growth and surveys their theory and computation, mainly for conservative systems while also treating broader applications.

  • LCEs characterize the asymptotic average growth or shrinking rate of small perturbations in dynamical systems.
  • The maximal LCE is commonly used to distinguish chaotic from regular orbits, with χ1 > 0 indicating chaos in the stated convention.
  • Its theory emphasizes Oseledec’s MET, while its computational coverage includes standard, SVD, and QR-based methods.
  • The report surveys both the theoretical framework and numerical techniques for computing maximal, few, and all LCEs.
  • The main scope is finite-dimensional autonomous Hamiltonian systems and symplectic maps on compact manifolds, excluding escapes to infinity.
  • Dissipative systems and time series are discussed briefly, extending beyond the report’s primary conservative-system focus.

2 Autonomous Hamiltonian systems and symplectic maps

This section develops Hamiltonian and symplectic dynamics, their tangent evolution, and numerical integration of the associated variational equations and tangent maps.

  • Autonomous Hamiltonian systems evolve in a 2N-dimensional phase space under Hamilton equations with constant Hamiltonian H = h.
  • Symplectic maps are area-preserving maps whose discrete states and iterations define the orbit evolution.
  • Variational equations and tangent map: Deviation vectors evolve in tangent spaces through the differential of the flow or through the tangent map of a symplectic map.
  • Numerical integration of variational equations: Hamiltonian variational equations must be integrated simultaneously with the reference orbit because their coefficients depend on that orbit.
  • Variational equations and tangent map: For continuous systems, the fundamental matrix satisfies the time-dependent linear equation Ẏ(t) = Df(x(t)) · Y(t), with Y(0) = I2N.
  • Numerical integration of variational equations: A numerical scheme samples the reference orbit at time steps and treats coefficients as constant over each interval, enabling explicit solution of the variational system.
  • Tangent dynamics of symplectic maps: For symplectic maps, deviation evolution likewise requires simultaneous iteration of the phase-space map and its orbit-dependent tangent map.

3 Historical introduction: The early days of LCEs

Early numerical studies connected the divergence of nearby trajectories with chaotic motion and later recognized this divergence as an estimator of the maximal LCE. Variational equations and orthonormalized deviation vectors then improved computation from the maximal exponent to multiple or complete LCE spectra.

  • LCEs quantify the average asymptotic growth or shrinking rate of small perturbations and provide quantitative measures of sensitivity to initial conditions.
  • Early orbit-divergence studies found approximately linear distance growth in regular regions and exponential growth in chaotic regions.These studies evolved pairs of trajectories with initial phase-space distances of about 10^-7–10^-6.
  • The quantity X1(t), derived from the distance between initially close orbits, was identified as an estimator of the maximal LCE as t →∞.For regular orbits, X1(t) tends to zero following a power law proportional to t^-1; for chaotic orbits, it tends to nonzero values.
  • Using variational equations instead of simultaneously integrating two nearby orbits enabled larger integration steps and removed the need to choose an initial separation.
  • Theoretical and numerical methods based on multiple deviation vectors and Gram–Schmidt orthonormalization enabled computation of some or all LCEs.Benettin et al. established the framework and applied it to complete spectra in several Hamiltonian systems, including four- and six-dimensional maps.

4 Lyapunov Characteristic Exponents: Theoretical treatment

The section develops the theoretical definition of Lyapunov Characteristic Exponents and the multiplicative-cocycle framework underlying their existence and computation. It relates vector and subspace growth to the LCE spectrum and describes how generic initial conditions recover maximal exponents and how orbit type affects the spectrum.

  • Theoretical framework: Oseledec’s multiplicative ergodic framework provides the theoretical basis for the existence and numerical evaluation of LCEs.The presentation follows Oseledec and Benettin et al. and applies the results to general matrix functions containing multiplicative cocycles.
  • Spectrum and orbit types: For the illustrated 3D Hamiltonian system, chaotic orbits have χ1 > χ2 nonzero while χ3 = 0, whereas regular orbits have vanishing LCEs.The zero exponent is associated with the flow direction in Hamiltonian systems.
  • Definitions and tangent dynamics: LCEs measure asymptotic exponential growth or shrinking rates of deviations under the tangent dynamics.The tangent map evolves deviation vectors through a matrix function forming a multiplicative cocycle.
  • Definitions and tangent dynamics: The p-LCE is defined from the long-time logarithmic growth rate of a p-dimensional parallelepiped volume.The volume is computed from the wedge product of p evolved deviation vectors and depends only on their spanned subspace.
  • Spectrum and generic directions: A random initial deviation vector generically yields the maximal LCE because its first nonzero component in the Lyapunov filtration eventually dominates.More specifically chosen vectors can target lower exponents, but numerical errors ultimately tend to restore the maximal-exponent computation.
  • Spectrum and orbit types: LCEs are constant across a connected chaotic domain for nonperiodic orbits, while unstable periodic orbits generally have different, orbit-specific spectra.Periodic orbits do not densely visit the whole chaotic domain and therefore do not characterize its constant nonperiodic spectrum.

5 The maximal LCE

The maximal Lyapunov characteristic exponent (mLCE) quantifies the asymptotic growth of nearby perturbations and serves as a chaos indicator. Its numerical computation evolves deviation vectors through variational dynamics while periodically renormalizing them to avoid overflow, and distinguishes regular from chaotic behavior through the long-time behavior of X1(t).

  • 5.1 Computation of the mLCE: For regular orbits, χ1 = 0, whereas chaotic orbits have χ1 > 0, indicating exponential divergence of nearby orbits.The mLCE has also been used as a physical time-scale indicator through its inverse, the Lyapunov time.
  • 5.2 The numerical algorithm: Direct propagation can cause numerical overflow because deviation-vector norms grow exponentially on chaotic orbits.The algorithm exploits linearity and composition of the tangent evolution by dividing time into intervals of length τ and renormalizing after each interval.
  • 5.1 Computation of the mLCE: The mLCE is obtained from the long-time limit of X1(t), equivalently from the mean of local stretching numbers along the orbit.Each local coefficient αi measures deviation-vector expansion over an interval τ, with ln αi/τ called the stretching number.
  • 5.2 The numerical algorithm: At every interval τ, the evolved deviation vector is replaced by a unit vector with the same direction, while its norm αi contributes to the estimate of χ1.This procedure is applied iteratively alongside the orbit evolution for Hamiltonian flows or symplectic maps until a good approximation is reached.
  • 5.3 Behavior of X1(t) for regular and chaotic orbits: For regular orbits, the deviation-vector norm grows almost linearly, so X1(t) decreases as O(ln t/t) and tends to zero like t−1.This behavior follows from constant action coordinates and linearly increasing angle coordinates on regular tori.
  • 5.3 Behavior of X1(t) for regular and chaotic orbits: For chaotic orbits, X1(t) is initially irregular and stabilizes only at large t toward the positive value χ1; smaller χ1 requires longer convergence times.Examples include χ1 ≈ 8·10−3 and χ1 ≈ 1.6 · 10−7, both compared with regular-orbit behavior proportional to t−1.

6 Computation of the spectrum of LCEs

The LCE spectrum provides information beyond the maximal exponent, including dynamical structure, entropy, and dimension-related quantities. The section develops volume-growth foundations and numerical procedures, especially the standard method and its QR interpretation, while noting scope-specific numerical limitations.

  • The LCE spectrum supplies information on underlying dynamics and statistical properties, and can measure fractal dimensions of strange attractors in dissipative systems.
  • Different spectra provide evidence that chaotic orbits belong to different noncommunicating chaotic regions, while similar spectra indicate possible membership in a connected region.
  • The p-mLCE is obtained from the long-time growth rate of p-dimensional volumes generated by p linearly independent deviation vectors, equaling the sum of the p largest 1-LCEs.
  • 6.3 Connection between the standard method and the QR decomposition: The standard method is practically a QR decomposition procedure and can compute either part or the whole LCE spectrum.
  • Computing the complete spectrum of LCEs: For continuous QR computation of only part of the spectrum, usual numerical integration can fail to preserve Q's orthogonality because the governing matrix is not skew-symmetric.

7 Chaos detection techniques

The section surveys numerical indicators that distinguish regular from chaotic orbits, emphasizing FLI and GALI as faster alternatives or complements to mLCE computation. It also notes that finite-time chaos indicators depend on norms or coordinates, unlike asymptotic LCEs.

  • Motivation: Poincaré sections and phase-space plots become difficult to interpret in multidimensional systems, motivating fast numerical chaos indicators.Regular and chaotic motion can be visually distinguished in low-dimensional systems, but higher-dimensional projections are often complicated and misleading.
  • Motivation: The mLCE χ1 indicates chaos when χ1 > 0, but estimating its asymptotic limit can require huge computation times for sticky orbits.The difficulty is especially pronounced when chaotic trajectories remain near regular behavior for long intervals.
  • Fast Lyapunov indicator: FLI tracks the current maximum norm of a deviation vector rather than the asymptotic mean stretching, enabling faster regular-chaotic discrimination.Regular orbits produce relatively small FLI values, whereas chaotic orbits produce very large values through exponential deviation growth.
  • Generalized alignment index: GALI_p evolves multiple deviation vectors and tends exponentially to zero for chaotic orbits, while regular orbits show constant or power-law behavior depending on p and torus dimension.For regular motion, GALI_p remains practically constant when p ≤ N and decays by a power law when p > N.
  • Generalized alignment index: GALI_p exploits the alignment that complicates many-LCE computation, providing detailed local-dynamics information and supporting torus-dimensionality estimates.Unlike successive orthonormalization methods, GALI uses the tendency of deviation vectors to become dependent as a diagnostic signal.
  • Caveat: Finite-time chaos indicators depend on the chosen norm or coordinates, whereas the asymptotic LCE spectrum does not.This coordinate dependence affects quantitative finite-time values such as X1(t).

8 LCEs of dissipative systems and time series

The report extends LCE theory and numerical methods beyond conservative systems to dissipative dynamics and observed time series. For time series, phase-space reconstruction and neighbor-based methods address the absence of governing equations, but limited data constrain precision.

  • Scope: LCE theory and evaluation techniques apply to general dynamical systems, although the report focuses mainly on Hamiltonian systems and symplectic maps.Dissipative systems and time series are treated briefly as complementary cases.
  • Dissipative systems: Dissipative dynamics contract phase-space volume onto lower-dimensional attractors and therefore include at least one negative LCE.Regular dissipative flows may have all negative exponents at fixed points or several zero exponents on quasiperiodic tori.
  • Dissipative systems: Strange attractors have one or more positive LCEs, corresponding to repeated stretching and folding that produces chaotic motion on a bounded attractor.The folding process is required to reconcile exponential expansion with bounded motion.
  • Dissipative systems: The methods for computing the maximal, partial, and complete LCE spectra can also be applied to dissipative systems.Many of these techniques were initially used in dissipative-model studies.
  • Time series: Time-series LCE estimation is difficult because experiments usually provide observations without the nonlinear equations governing the system.Phase-space reconstruction with delay coordinates creates a hypothetical orbit in a reconstructed d-dimensional space.
  • Time series: Neighbor-based methods estimate the maximal LCE from reconstructed time series, while tangent-space methods target the whole spectrum.The direct method follows nearest-neighbor separations, whereas the tangent-space method numerically determines local evolution matrices.
  • Time series: The direct time-series method may be imprecise because reliable deviation-vector replacement requires data conditions that are often infeasible with limited observations.This limitation is explicitly attributed to the restricted number of data points.

A Exterior algebra and wedge product: Some basic notions

The section introduces exterior algebra and wedge products as tools for representing oriented volumes formed by multiple vectors. It develops bases, inner products, norms, and determinant-based representations of k-vectors.

  • Basic notions: The wedge product ∧ is the exterior product on an M-dimensional real vector space and generates k-vectors from k vectors.The k-th exterior power Λ^k(V) is the subspace generated by all such products.
  • Bases: Lexicographically ordered wedge products of orthonormal basis vectors form an orthonormal basis of Λ^k(V).The basis contains all index combinations satisfying 1 ≤ i1 < i2 < ··· < ik ≤ M.
  • Basic notions: A wedge product of linearly dependent vectors is zero, while a product of linearly independent vectors is a decomposable k-vector.Not every k-vector is decomposable; the section gives a non-decomposable 2-vector example in Λ^2(R^4).
  • Coordinate representation: For decomposable k-vectors, the coefficients in the wedge expansion are minors of the matrix whose columns contain the generating vectors’ coordinates.The determinant of each selected k×k submatrix supplies the corresponding basis coefficient.
  • Inner product and norm: The induced inner product on Λ^k(V) is expressed through a determinant of pairwise vector inner products and defines a norm on k-vectors.The induced basis inner product satisfies ⟨ω_i,ω_j⟩_k = δ_ij.
  • Geometric meaning: The norm of a decomposable k-vector equals the volume of the k-parallelogram spanned by its generating vectors and is invariant under orthonormal basis changes.The basis invariance follows from the orthogonality of the transformation matrix.

A.1 An illustrative example

The illustrative example specializes the exterior-algebra construction to R^4, listing bases for Λ^2(R^4), Λ^3(R^4), and Λ^4(R^4). It also connects wedge-product norms with Gram-matrix determinants for two, three, and four vectors.

  • Two-vectors: For V = R^4, Λ^2(R^4) has dimension 6 with the lexicographically ordered basis e1∧e2, e1∧e3, e1∧e4, e2∧e3, e2∧e4, e3∧e4.These six basis vectors are explicitly listed as ω1 through ω6.
  • Four-vectors: Λ^4(R^4) is one-dimensional and is spanned by the wedge product of the four standard basis vectors.The example then considers four linearly independent vectors and their coordinate matrix.
  • Norm calculations: The example uses the usual Euclidean norm for vectors while applying the exterior-algebra norm to the resulting multivectors.This distinguishes the vector norm notation from the induced norm used for wedge products.
  • Norm calculations: The norms of the 3-vector and 4-vector wedges are computed using Gram-type matrices of pairwise inner products.The example presents the corresponding 3×3 and 4×4 determinant expressions.
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