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Analysis and Mathematical Physics

, Volume 9, Issue 1, pp 523–529 | Cite as

Computing the Douady–Earle extension using Kuramoto oscillators

  • Vladimir JaćimovićEmail author
Article

Abstract

We demonstrate that the Douady–Earle extension can be computed by solving the specific set of ODE’s. This system of ODE’s has several interpretations in Mathematical Physics, such as Kuramoto model of coupled oscillators or Josephson junction arrays. This method emphasizes the key role of conformal barycenter in some theories of Mathematical Physics. On the other hand, it indicates that variations of Kuramoto model might be used in different problems of computational quasiconformal geometry. The idea can be extended to compute the D–E extension in higher dimensions as well.

Keywords

Conformal barycenter Douady–Earle extension Coupled oscillators Kuramoto model 

Notes

Acknowledgements

The author wishes to thank his colleague prof. David Kalaj for valuable discussions.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Douady, A., Earle, C.J.: Conformally natural extensions of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abikoff, W., Ye, T.: Computing the Douady–Earle extension. Contemp. Math. 211, 1–8 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abikoff, W.: Conformal barycenters and the Douady–Earle extension—a discrete dynamical approach. J. d’Analyse Math. 86, 221–234 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Abikoff, W., Earle, C.J., Mitra, S.: Barycentric extensions of monotone maps on the circle. In: In Abikoff, W., Haas, A. (eds.)Tradition of Ahlfors and Bers. Contemp. Math, vol. 355, pp. 1–20 (2004)Google Scholar
  5. 5.
    Marvel, S., Mirollo, R., Strogatz, S.H.: Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action. Chaos 19, 043104 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Watanabe, S., Strogatz, S.H.: Constants of motion for superconducting Josephson arrays. Physica D 74, 197–253 (1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  8. 8.
    Tanaka, T.: Solvable model of the collective motion of heterogeneous particles interacting on a sphere. New J. Phys. 16, 023016 (2014)CrossRefGoogle Scholar
  9. 9.
    Chen, B., Engelbrecht, J.R., Mirollo, R.: Hyperbolic geometry of Kuramoto oscillator networks. J. Phys. A Math. Theor. 50, 355101 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Natural Sciences and MathematicsUniversity of MontenegroPodgoricaMontenegro

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