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Coupled Nonautonomous Oscillators

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Nonautonomous Dynamical Systems in the Life Sciences

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2102))

Abstract

First, we introduce nonautonomous oscillator—a self-sustained oscillator subject to external perturbation and then expand our formalism to two and many coupled oscillators . Then, we elaborate the Kuramoto model of ensembles of coupled oscillators and generalise it for time-varying couplings. Using the recently introduced Ott-Antonsen ansatz we show that such ensembles of oscillators can be solved analytically. This opens up a whole new area where one can model a virtual physiological human by networks of networks of nonautonomous oscillators. We then briefly discuss current methods to treat the coupled nonautonomous oscillators in an inverse problem and argue that they are usually considered as stochastic processes rather than deterministic. We now point to novel methods suitable for reconstructing nonautonomous dynamics and the recently expanded Bayesian method in particular. We illustrate our new results by presenting data from a real living system by studying time-dependent coupling functions between the cardiac and respiratory rhythms and their change with age. We show that the well known reduction of the variability of cardiac instantaneous frequency is mainly on account of reduced influence of the respiration to the heart and moreover the reduced variability of this influence. In other words, we have shown that the cardiac function becomes more autonomous with age, pointing out that nonautonomicity and the ability to maintain stability far from thermodynamic equilibrium are essential for life.

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Notes

  1. 1.

    The authors contributed equally to this work and have therefore decided for an alphabetical order.

  2. 2.

    The following statement holds also for higher frequency ratios in the form \(\psi = n\phi _{2} - m\phi _{1}\) where n and m are integer numbers.

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Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council (UK) [Grant No. EP/100999X1]. Our grateful thanks are due to A. Duggento, D. Iatsenko, P.V.E. McClintock and Y. Suprunenko for many useful discussions.

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Authors and Affiliations

  1. Physics Department, Lancaster University, Lancaster, UK

    Philip T. Clemson, Spase Petkoski, Tomislav Stankovski & Aneta Stefanovska

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  1. Philip T. Clemson
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  2. Spase Petkoski
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Correspondence to Aneta Stefanovska .

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  1. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Peter E. Kloeden

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt am Wörthersee, Austria

    Christian Pötzsche

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Clemson, P.T., Petkoski, S., Stankovski, T., Stefanovska, A. (2013). Coupled Nonautonomous Oscillators. In: Kloeden, P., Pötzsche, C. (eds) Nonautonomous Dynamical Systems in the Life Sciences. Lecture Notes in Mathematics(), vol 2102. Springer, Cham. https://doi.org/10.1007/978-3-319-03080-7_5

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