Nonlinear Dynamics

, Volume 85, Issue 1, pp 195–201 | Cite as

Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

  • N. V. Kuznetsov
  • T. A. Alexeeva
  • G. A. Leonov
Original Paper


Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work, the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance of Lyapunov exponents for regular and irregular linearizations under the change of coordinates is demonstrated.


Lyapunov exponent Lyapunov characteristic exponent  Lyapunov dimension of attractor Time-varying linearization  Regular and irregular linearization Diffeomorphism 



This work was supported by Russian Scientific Foundation project 14-21-00041 and Saint-Petersburg State University.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • N. V. Kuznetsov
    • 1
    • 2
  • T. A. Alexeeva
    • 3
  • G. A. Leonov
    • 1
    • 4
  1. 1.Saint-Petersburg State UniversitySaint PetersburgRussia
  2. 2.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  3. 3.National Research University Higher School of EconomicsSaint PetersburgRussia
  4. 4.Institute of Problems of Mechanical Engineering RASSaint PetersburgRussia

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