Journal of Nonlinear Science

, Volume 22, Issue 5, pp 727–762 | Cite as

Theory and Computation of Covariant Lyapunov Vectors

  • Pavel V. KuptsovEmail author
  • Ulrich Parlitz


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.


Covariant Lyapunov vectors Characteristic Lyapunov vectors Forward and backward Lyapunov vectors Lyapunov exponents Lyapunov analysis Tangent space High-dimensional chaos 



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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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Instrumentation EngineeringSaratov State Technical UniversitySaratovRussia
  2. 2.Biomedical Physics GroupMax Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  3. 3.Institute for Nonlinear DynamicsGeorg-August-Universität GöttingenGöttingenGermany

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