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Bistability of rotational modes in a system of coupled pendulums

  • Nonlinear Dynamics & Mobile Robotics
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Abstract

The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.

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Authors and Affiliations

  1. Lobachevsky State University of Nizhni Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    Lev A. Smirnov, Alexey K. Kryukov, Grigory V. Osipov & Jürgen Kurths

  2. Institute of Applied Physics of the Russian Academy of Sciences, ul. Ulyanova 46, Nizhny Novgorod, 603950, Russia

    Lev A. Smirnov & Jürgen Kurths

  3. Potsdam Institute for Climate Impact Research, Telegrafenberg, Potsdam, 14473, Germany

    Jürgen Kurths

  4. Humboldt-Universitat zu Berlin, Unter den Linden 6, Berlin, 10099, Germany

    Jürgen Kurths

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  1. Lev A. Smirnov
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Correspondence to Lev A. Smirnov.

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Smirnov, L.A., Kryukov, A.K., Osipov, G.V. et al. Bistability of rotational modes in a system of coupled pendulums. Regul. Chaot. Dyn. 21, 849–861 (2016). https://doi.org/10.1134/S156035471607008X

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  • DOI: https://doi.org/10.1134/S156035471607008X

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