About Lyapunov Exponents Identification for Systems with Periodic Coefficients

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Nikolay N. Karabutov 1,*

1. Moscow Technological University (MIREA)/ Department of Problems Control, Moscow, Russia

* Corresponding author.

DOI: https://doi.org/10.5815/ijisa.2018.11.01

Received: 8 May 2018 / Revised: 11 Jun. 2018 / Accepted: 15 Jul. 2018 / Published: 8 Nov. 2018

Index Terms

Dynamic systems with periodic coeffi-cients, Lyapunov exponents, framework, histogram


Lyapunov exponents (LE) identification prob-lem of dynamic systems with periodic coefficients is con-sidered under uncertainty. LE identification is based on the analysis of framework special class describing dy-namics of their change. Upper bound for the smallest LE and mobility limit for the large LE are obtained and the indicator set of the system is determined. The graphics criteria based on the analysis of framework special class features are proposed for an adequacy estimation of obtained LE estimations. The histogram method is applied to check for obtained estimation set. We show that the dynamic system can have the LE set.

Cite This Paper

Nikolay Karabutov, "About Lyapunov Exponents Identification for Systems with Periodic Coefficients", International Journal of Intelligent Systems and Appli-cations(IJISA), Vol.10, No.11, pp.1-10, 2018. DOI:10.5815/ijisa.2018.11.01


[1]K. Thamilmaran, D.V. Senthilkumar, A. Venkatesan, M. Lakshmanan, “Experimental realization of strange non-chaotic attractors in a quasiperiodically forced electronic circuit,” Physical Review E, vol. 74, no. 9, pp. 036205, 2006.
[2]R. Porcher, G., “Thomas Estimating Lyapunov exponents in biomedical time series,” Physical Review E, vol. 64, no. 1, pp. 010902(R), 2001.
[3]A. Goshvarpour, A. Goshvarpour, “Chaotic Behavior of Heart Rate Signals during Chi and Kundalini Meditation.” International Journal of Image, Graphics and Signal Pro-cessing, vol. 4, no. 2, pp. 23-29, 2012.
[4]A. Goshvarpour, A. Abbasi and A. Goshvarpour, “Non-linear Evaluation of Electroencephalogram Signals in Dif-ferent Sleep Stages in Apnea Episodes,” International journal of intelligent systems and applications, vol. 5, no. 10, pp. 68-73, 2013.
[5]J.A. HoƂyst, K. Urbanowicz, “Chaos control in economical model by time-delayed feedback method,” Physica A: Statistical Mechanics and its Applications, vol. 287, no. 3–4, pp. 587–598, 2000.
[6]W.M. Macek, S. Redaelli, “Estimation of the entropy of the solar wind flow,” Physical Review E, vol. 62, no. 5, pp. 6496–6504, 2000.
[7]Ch. Skokos, “The Lyapunov Characteristic Exponents and Their Computation,” Lect. Notes Phys, vol. 790, pp. 63–135, 2010.
[8]R. Gencay, W.D. Dechert, “An algorithm for the n Lya-punov exponents of an n-dimensional unknown dynamical system,” Physica D, vol. 59, pp. 142-157, 1992.
[9]F. Takens, “Detecting strange attractors in turbulence,” Dynamical Systems and Turbulence. Lecture Notes in Mathematics /Eds D. A. Rand, L.-S. Young. Berlin: Springer-Verlag, vol. 898. pp. 366–381, 1980.
[10]A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, “Deter-mining Lyapunov exponents from a time series,” Physica 16D, no. 16, pp. 285–301, 1985.
[11]V.A. Bazhenov, O.S. Pogorelova, T.G. Postnikova, “Lya-punov exponents estimation for strongly nonlinear non-smooth discontinuous vibroimpact system,” Strength of Materials and Theory of Structures, 2017, is. 99, pp. 90 – 105.
[12]A.V. Bespalov, N.D. Polyakhov, “Comparative analysis of methods for estimating the first Lyapunov exponent,” Modern problems of science and education, no 6, 2016.
[13]V.A. Golovko, "Neural network methods of chaotic pro-cesses processing," In Scientific session of MEPhI-2005. VII All-Russian scientific and technical Neuroinformation scientist(Neuroinformatics)-2005 conference "Neuroin-formatics 2005": Lectures on neuroinformatics. Moscow: MEPhI, pp. 43-88, 2005.
[14]Y.A. Perederiy, “Method for calculation of lyapunov ex-ponents spectrum from data series,” Izvestiya VUZ. Applied nonlinear dynamics, is. 20, no. 1, pp. 99-104, 2012.
[15]G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for com-puting all of them,” Pt. I: Theory. Pt. II: Numerical appli-cations, Meccanica, vol. 15, pp. 9–30, 1980.
[16]O. Moskalenko, A. A. Koronovskii, A.E, Hramov, “Lya-punov exponent corresponding to enslaved phase dynamics: Estimation from time series,” Physical review E 92, 2015, 012913.
[17]Cvitanovi´c P., Artuso R., Mainieri R., Tanner G., Vattay G., Chaos: Classical and Quantum, ChaosBook.org ver-sion16.0, 2017.
[18]V.V. Filatov, “Structural characteristics of geophysical fields anomalies and their use in forecasting,” Geophysics, no. 4(16), pp. 34-41, 2013.
[19]B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemyt-sky, Theory of Lyapunov indexes and its application to stability problems. Moscow: Nauka, 1966.
[20]N. N. Karabutov, Frameworks in problems of identification: Design and analysis. Moscow: URSS/Lenand. 2018 (in Russian).
[21]G. Bohr. Almost periodic functions. Moscow: Librocom, 2009.