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Advances in Dynamics, Patterns, Cognition pp 13–28Cite as

Lorenz Type Attractor in Electronic Parametric Generator and Its Transformation Outside the Parametric Resonance

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Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 20))

Abstract

For parametric generator composed of three LC circuits and a quadratic nonlinear element on the basis of the operational amplifier and analog multiplier the equations for the interacting mode amplitudes are derived. At frequencies providing the parametric resonance precisely, the problem reduces to a system of three differential equations of the first order for the amplitudes with attractor of Lorenz type. In the presence of the detuning not only the dynamics of amplitudes are relevant, but also the phase dynamics. Attractor must be considered now in the six-dimensional phase space, and in contrast to the Lorenz attractor it is not quasi-hyperbolic.

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Notes

  1. 1.

    Lyapunov exponents’ calculations were performed on time intervals of duration of 50,000 with the counting of the average values and standard deviations on 20 samples. As an error, the standard deviations are indicated.

  2. 2.

    The lack of a perfect match in the parameters for Fig. 2.9 in comparison to Fig. 2.8 is due to the approximate nature of description in terms of slow amplitudes.

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Authors and Affiliations

  1. Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov, 410019, Russia

    Sergey P. Kuznetsov

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  1. Sergey P. Kuznetsov
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Correspondence to Sergey P. Kuznetsov .

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Editors and Affiliations

  1. Pennsylvania State University, University Park, Pennsylvania, USA

    Igor S. Aranson

  2. Institute for Physics and Astronomy, University of Potsdam, Potsdam, Germany

    Arkady Pikovsky

  3. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Nikolai F. Rulkov

  4. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Lev S. Tsimring

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Kuznetsov, S.P. (2017). Lorenz Type Attractor in Electronic Parametric Generator and Its Transformation Outside the Parametric Resonance. In: Aranson, I., Pikovsky, A., Rulkov, N., Tsimring, L. (eds) Advances in Dynamics, Patterns, Cognition. Nonlinear Systems and Complexity, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-53673-6_2

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  • DOI: https://doi.org/10.1007/978-3-319-53673-6_2

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