Acta Scientific Computer Sciences

Research Article Volume 4 Issue 4

Estimation of Lyapunov Exponents for Systems with Periodic Coefficients On-base Structure Approach

Nikolay Karabutov*

Department of Control Problems, MIREA-Russian Technological University, Moscow, Russia

*Corresponding Author: Nikolay Karabutov, Department of Control Problems, MIREA-Russian Technological University, Moscow, Russia.

Received: November 02, 2021; Published: March 28, 2022


We propose an approach to Lyapunov exponents (LE) identification. It bases on the analysis of geometric frameworks (GF) describing the dynamics of the LE change. We obtain the upper bound for the smallest LE and mobility limit for the large LE and the indicator set for the system with periodic coefficients. Graphical criteria (GC) propose to assess of adequacy the obtained LE. GC is based on the proposed histograms method and is applied to assess LE adequacy. We show that the dynamic system has the LE set. Identifiability conditions of LE are obtained

Keywords: Dynamic Systems with Periodic Coefficients; Lyapunov Exponents; Framework; Histogram; Almost Periodic Function; Identifiability


  1. Thamilmaran K., et al. “Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit”. Physical Review E 74 (2006): 036205.
  2. Porcher R and Thomas G. “Estimating Lyapunov exponents in biomedical time series”. Physical Review E 7 (2001): 010902.
  3. Goshvarpour A and Goshvarpour A. “Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation”. International Journal of Image, Graphics and Signal Processing 2 (2012): 23-29.
  4. Goshvarpour A., et al. “Nonlinear Evaluation of Electroencephalogram Signals in Different Sleep Stages in Apnea Episodes”. International Journal of Intelligent Systems and Applications 10 (2013): 68-73.
  5. Hołyst JA and Urbanowicz K. “Chaos control in economical model by time-delayed feedback method”. Physica A: Statistical Mechanics and its Applications 3-4 (2000): 587-598.
  6. Macek WM and Redaelli S. “Estimation of the entropy of the solar wind flow”. Physical Review E 5 (2000): 6496-6504.
  7. Ch Skokos. “The Lyapunov Characteristic Exponents and Their Computation”. Lecture Notes in Physics 790 (2010): 63-135.
  8. Gencay R and Dechert WD. “An algorithm for the n Lyapunov exponents of an n-dimensional unknown dynamical system”. Physica D 59 (1992): 142-157.
  9. Takens F. “Detecting strange attractors in turbulence”. Dynamical Systems and Turbulence. Lecture Notes in Mathematics/Editions D. A. Rand, L.-S. Young. Berlin: Springer-Verlag 898 (1980): 366-381.
  10. Wolf A., et al. “Determining Lyapunov exponents from a time series”. Physica 16D 16 (1985): 285-301.
  11. Bazhenov VA., et al. “Lyapunov exponents estimation for strongly nonlinear nonsmooth discontinuous vibroimpact system”. Strength of Materials and Theory of Structures 99 (2017): 90-105.
  12. Bespalov AV and Polyakhov ND. “Comparative analysis of methods for estimating the first Lyapunov exponent”. Modern Problems of Science and Education 6 (2016).
  13. Golovko VA. "Neural network methods of chaotic processes processing". In Scientific session of MEPhI-2005. VII All-Russian scientific and technical Neuroinformation scientist (Neuroinformatics)-2005 conference "Neuroinformatics 2005": Lectures on neuroinformatics. Moscow: MEPhI (2005): 43-88.
  14. Perederiy YA. “Method for calculation of lyapunov exponents spectrum from data series”. Izvestiya VUZ. Applied Nonlinear Dynamics 1 (2012): 99-104.
  15. Benettin G., et al. “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them”. Pt. I: Theory. Pt. II: Numerical applications, Meccanica 15 (1980): 9-30.
  16. Moskalenko O., et al. “Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series”. Physical Review E 92 (2015): 012913.
  17. Cvitanovi´c P., et al. “Chaos: Classical and Quantum”. version16.0 (2017).
  18. Filatov VV. “Structural characteristics of geophysical fields anomalies and their use in forecasting”. Geophysics 16 (2013): 34-41.
  19. Bylov BF., et al. “Theory of Lyapunov indexes and its application to stability problems”. Moscow: Nauka (1966).
  20. Karabutov NN. “Frameworks in problems of identification: Design and analysis”. Moscow: URSS/Lenand (2018).
  21. Bohr G. “Almost periodic functions”. Moscow: Librocom (2009).
  22. Karabutov N. “About Lyapunov Exponents Identification for Systems with Periodic Coefficients”. International Journal of Intelligent Systems and Applications 11 (2018): 1-10.
  23. Karabutov N. “Geometrical Frameworks in Identification Problem”. Intelligent Control and Automation 12 (2021): 17-43.


Citation: Nikolay Karabutov. “Estimation of Lyapunov Exponents for Systems with Periodic Coefficients On-base Structure Approach”. Acta Scientific Computer Sciences 4.4 (2022): 83-90.


Copyright: © 2022 Nikolay Karabutov. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.


Acceptance rate35%
Acceptance to publication20-30 days

Indexed In

News and Events

  • Certification for Review
    Acta Scientific certifies the Editors/reviewers for their review done towards the assigned articles of the respective journals.
  • Submission Timeline for Upcoming Issue
    The last date for submission of articles for regular Issues is July 10, 2022.
  • Publication Certificate
    Authors will be issued a "Publication Certificate" as a mark of appreciation for publishing their work.
  • Best Article of the Issue
    The Editors will elect one Best Article after each issue release. The authors of this article will be provided with a certificate of “Best Article of the Issue”.
  • Welcoming Article Submission
    Acta Scientific delightfully welcomes active researchers for submission of articles towards the upcoming issue of respective journals.
  • Contact US