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Theory and Computation of Covariant Lyapunov Vectors

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Abstract

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.

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Acknowledgements

The research leading to the results has received funding from the European Community’s Seventh Framework Programme FP7/2007–2013 under grant agreement No. HEALTH-F2-2009-241526, EUTrigTreat. P.V.K. acknowledges support from RFBR-DFG under Grant No. 08-02-91963.

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Authors and Affiliations

  1. Department of Instrumentation Engineering, Saratov State Technical University, Politekhnicheskaya 77, Saratov, 410054, Russia

    Pavel V. Kuptsov

  2. Biomedical Physics Group, Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077, Göttingen, Germany

    Ulrich Parlitz

  3. Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Am Faßberg 17, 37077, Göttingen, Germany

    Ulrich Parlitz

Authors
  1. Pavel V. Kuptsov
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  2. Ulrich Parlitz
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Correspondence to Pavel V. Kuptsov.

Additional information

Communicated by P. Newton.

Appendix: Pseudocode for the LU Method

Appendix: Pseudocode for the LU Method

Input: nclv, number of computed covariant Lyapunov vectors; nstore, number of trajectory points where the covariant vectors are computed; m, dimension of the tangent space; dt, time interval between orthogonalizations (normally, a multiple of time discretization step); nspend_att, nspend_fwd, nspend_bkw,steps to converge to the attractor, forward and backward vectors, respectively.

Subroutines: solve_bas(), solving of the basic system; solve_lin_fwd(), solve_lin_trp(), action of forward and transposed propagators, respectively (see Sect. 3); null_vect(), computing a null vector (in the case of multiple solutions, an arbitrary null vector can be taken); orthog(), QR-orthogonalization (matrix \(\mbox {\boldmath {$\mathrm {R}$}}\) is abandoned); transpose(), transpose of a matrix; random(), generate random matrix or vector; A.B, multiplication of matrices A and B.

Result: Gamma, array of nstore matrices m by nclv, whose columns are the covariant Lyapunov vectors

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Kuptsov, P.V., Parlitz, U. Theory and Computation of Covariant Lyapunov Vectors. J Nonlinear Sci 22, 727–762 (2012). https://doi.org/10.1007/s00332-012-9126-5

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