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

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Abstract

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.

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Notes

  1. In [42], these values are defined with the opposite sign and are called characteristic exponents.

  2. For example, for the fundamental matrix \(X(t)={\left( \begin{array}{c@{\quad }c} 1 &{} g(t)-g^{-1}(t) \\ 0 &{} 1 \\ \end{array} \right) }\) we have the following ordered values: \( {{\mathrm{LCE}}}_1^{o} = \mathrm{max}\big (\limsup _{t \rightarrow +\infty }\mathcal {X}(g(t))\), \(\limsup _{t \rightarrow +\infty }\mathcal {X}(g^{-1}(t))\big )\), \({{\mathrm{LCE}}}_2^{o} = 0; \mathrm{LE}_{1,2}^{o} = \mathrm{max, min} \big ( \limsup _{t \rightarrow +\infty }\mathcal {X}(g(t)), \limsup _{t \rightarrow +\infty }\mathcal {X}(g^{-1}(t)) \big ). \) Remark that here \(\mathcal {X}\) of the diagonal elements of X(t) do not coincide with \({{\mathrm{LCE}}}\) s and \(\mathrm{LE}\) s.

  3. This is not a dimension in a rigorous sense (see, e.g., [22, 26]); for example, in [35, Fig. 7;p.1439], a local B-attractor, which includes equilibria and separatrices, has \(\dim _L \approx 2.8\).

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Acknowledgments

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

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Authors and Affiliations

  1. Saint-Petersburg State University, Saint Petersburg, Russia

    N. V. Kuznetsov & G. A. Leonov

  2. Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    N. V. Kuznetsov

  3. National Research University Higher School of Economics, Saint Petersburg, Russia

    T. A. Alexeeva

  4. Institute of Problems of Mechanical Engineering RAS, Saint Petersburg, Russia

    G. A. Leonov

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Kuznetsov, N.V., Alexeeva, T.A. & Leonov, G.A. Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations. Nonlinear Dyn 85, 195–201 (2016). https://doi.org/10.1007/s11071-016-2678-4

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