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Chaos / Volume 19 / Issue 4 / REGULAR ARTICLES

Chaos 19, 043120 (2009); http://dx.doi.org/10.1063/1.3266924 (6 pages)

Synchronization and chaotic dynamics of coupled mechanical metronomes

Henning Ulrichs1, Andreas Mann1, and Ulrich Parlitz2

1I. Physikalisches Institut, University of Göttingen, D-37077 Göttingen, Friedrich-Hund-Platz 1, Göttingen, Germany
2III. Physikalisches Institut, University of Göttingen, D-37077 Göttingen, Friedrich-Hund-Platz 1, Göttingen, Germany

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(Received 17 August 2009; accepted 28 October 2009; published online 3 December 2009)

Synchronization scenarios of coupled mechanical metronomes are studied by means of numerical simulations showing the onset of synchronization for two, three, and 100 globally coupled metronomes in terms of Arnol’d tongues in parameter space and a Kuramoto transition as a function of coupling strength. Furthermore, we study the dynamics of metronomes where overturning is possible. In this case hyperchaotic dynamics associated with some diffusion process in configuration space is observed, indicating the potential complexity of metronome dynamics.

© 2009 American Institute of Physics

Lead Paragraph

Metronomes are often viewed as paradigm of strictly periodic motion. Physically a metronome is a nonlinear self-sustained oscillator whose natural frequency is precisely tuned for practical applications, for example, to provide exact timing for musical exercises. Since the state space of a single metronome is two-dimensional, asymptotically only periodic oscillations or steady states may occur. This situation changes qualitatively if two or more metronomes are coupled and higher dimensional (chaotic) attractors are possible. Furthermore, coupled metronomes may exhibit synchronization phenomena reminiscent of Huygens’ famous pendulum clocks. If two metronomes are coupled they exhibit typical phase-locking phenomena if their individual (uncoupled) frequencies almost coincide or are rationally related. For large ensembles of metronomes a collective transition to synchronization occurs in terms of a so-called Kuramoto transition. And, last but not least, it turns out that coupled metronomes may exhibit high-dimensional chaotic dynamics, in particular if overturnings are possible. These findings and phenomena show that the dynamics of coupled metronomes is very rich and may also serve as paradigm of synchronization and complex dynamics.

Article Outline

  1. INTRODUCTION
  2. FORMULATION OF THE PROBLEM
  3. COUPLING OF TWO METRONOMES
  4. COUPLING OF THREE METRONOMES: PHASE DIFFUSION
  5. COUPLING OF N METRONOMES: KURAMOTO TRANSITION
  6. CONCLUSION

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KEYWORDS and PACS

Keywords

chaos, diffusion, nonlinear dynamical systems, numerical analysis, synchronisation

PACS

  • 05.45.Xt

    Synchronization; coupled oscillators

  • 05.60.-k

    Transport processes

  • 05.45.Pq

    Numerical simulations of chaotic systems

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3266924

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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    References

    B. Georges, J. Grollier, V. Cros, and A. Fert, Appl. Phys. Lett. 92, 232504 (2008)APPLAB000092000023232504000001.


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