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Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria

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Abstract

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

  1. Here, we mean the direction being parallel to the OZ axis.

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Acknowledgements

V.V.S. and A.S.Z. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer—163436311-SFB-910. V.V.S. also acknowledges support by the Russian Science Foundation (Project No. 22-72-00038).

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Authors and Affiliations

  1. Saratov State University, Astrakhanskaya str. 83, Saratov, Russia, 410012

    lvan A. Korneev, Andrei V. Slepnev, Tatiana E. Vadivasova & Vladimir V. Semenov

  2. Technische Universität Berlin, Hardenbergstraße 36, 10623, Berlin, Germany

    Anna S. Zakharova & Vladimir V. Semenov

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  1. lvan A. Korneev
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  2. Andrei V. Slepnev
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  3. Anna S. Zakharova
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  4. Tatiana E. Vadivasova
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  5. Vladimir V. Semenov
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Correspondence to Vladimir V. Semenov.

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Korneev, l.A., Slepnev, A.V., Zakharova, A.S. et al. Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria. Nonlinear Dyn 111, 1235–1243 (2023). https://doi.org/10.1007/s11071-022-07905-6

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