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Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy

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Abstract

In this paper we summarize and substantiate the relative metric entropy approach introduced in our previous papers [1, 2]. Using this approach we study the mixing influence of noise on both regular and chaotic systems. We show that the synchronization phenomenon as well as stochastic resonance decrease, the degree of mixing is caused by white Gaussian noise.

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References

  1. Anishchenko, V. S. and Astakhov, S. V., Relative Entropy as the Degree of Mixing of Systems in the Presence of Noise, Tech. Phys. Lett., 2007, vol. 33, no. 21, pp. 1–8 (Russian).

    Google Scholar 

  2. Anishchenko, V. S. and Astakhov, S.V., Relative Kolmogorov Entropy of a Chaotic System in the Presence of Noise, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 18, no. 9, pp. 2851–2855.

  3. Shannon, C.E., A Mathematical Theory of Communication, Bell System Tech. J., 1948, vol. 27, pp. 379–423, 623–656.

    MATH  MathSciNet  Google Scholar 

  4. Kolmogorov, A.N., Entropy per Unit Time as a Metric Invariant of Automorphisms, Dokl. Akad. Nauk SSSR, 1959, vol. 124, pp. 754–755.

    MATH  MathSciNet  Google Scholar 

  5. Pesin, Ya.B., Characteristic Ljapunov Exponents, and Smooth Ergodic Theory, Uspehi Mat. Nauk, 1977, vol. 32, no. 4(196), pp. 55–112, 287 [Russian Math. Surveys, 1977, vol. 32, no. 4, pp. 55–114].

    MathSciNet  Google Scholar 

  6. Faure, Ph. and Korn, H., A New Method to Estimate the Kolmogorov Entropy from Recurrence Plots: Its Application to Neuronal Signals, Phys. D, 1998, vol. 122, pp. 265–279.

    Article  Google Scholar 

  7. Marwan, N., Romano, M.Carmen, Thiel, M., and Kurths, J., Recurrence Plots for the Analysis of Complex Systems, Phys. Rep., 2007, vol. 438, nos. 5–6, pp. 237–329.

    Article  MathSciNet  Google Scholar 

  8. Grassberger, P. and Procaccia, I., Characterization of Strange Attractors, Phys. Rev. Lett., 1983, vol. 50, no. 5, pp. 346–349.

    Article  MathSciNet  Google Scholar 

  9. Grassberger, P. and Procaccia, I., Estimation of the Kolmogorov Entropy from a Chaotic Signal, Phys. Rev. A, 1983, vol. 28, pp. 2591–2593.

    Article  Google Scholar 

  10. Rössler, O.E., An Equation for Continuous Chaos, Phys. Lett. A, 1976, vol. 57, pp. 397–398.

    Article  Google Scholar 

  11. V.S. Anishchenko, Dynamical Chaos. Models and Experiments: Appearance, Routes and Structure of Chaos in Simple Dynamical Systems, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 8, River Edge, NJ: World Sci. Publ., 1995.

    Google Scholar 

  12. Shimansky-Geier, L. and Herzel, H., Positive Lyapunov-Exponents in the Kramers-Oscillator, J. Stat. Phys., 1993, vol. 70, pp. 141–147.

    Article  Google Scholar 

  13. Anishchenko, V., Astakhov, S., and Vadivasova, T., Phase Dynamics of Two Coupled Oscillators under External Periodic Force, Europhys. Lett., 2009, vol. 86, 30003.

    Article  Google Scholar 

  14. V. S. Anishchenko, S. V. Astakhov, T. E. Vadivasova and A. V. Feoktistov, Numerical and experimental study of external synchronization of two-frequency oscillations, Rus. J. Nonlin. Dyn., 2009, vol. 5, no. 2, pp. 237–252.

    Google Scholar 

  15. Anishchenko, V. S., Neiman, A. B., Moss, F., and Shimansky-Geier, L., Stochastic Resonance: Noise-Enhanced Order, Phys. Usp., 1999, vol. 42, pp. 7–36.

    Article  Google Scholar 

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Anishchenko, V.S., Astakhov, S.V. & Vadivasova, T.E. Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy. Regul. Chaot. Dyn. 15, 261–273 (2010). https://doi.org/10.1134/S1560354710020139

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  • DOI: https://doi.org/10.1134/S1560354710020139

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