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Phase locking control in the Circle Map

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Abstract

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.

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References

  1. Ch. Huygens: Horologium Oscillatorium. Apud. F. Muguet, Parisiss, France (1673)

  2. Pikovsky, A., Rosenblum, M., Kurths, J.: Syncrhonization – A Universal Concept in Nonlinear Sciences. Cambridge, United Kingdom (2001)

  3. Schafer, C., Rosenblum, M.G., Kurths, J., Abel, H.H.: Heartbeat synchronized with ventilation. Nature 392(6673), 239–240 (1998)

    Article  Google Scholar 

  4. Buck, J.: Synchronous rhythmic flashing of fireflies. Quart. Rev. Biol. 63, 265–288 (1988)

    Article  Google Scholar 

  5. Battin, R. H. An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, Reston, VA (1999)

  6. Ecke, R. E., Farmer, J. D., Umberger., D. K.: Scaling of the Arnold tonges. Nonlinearity 2, 175–196 (1989)

    Article  MathSciNet  Google Scholar 

  7. Arnold, V.I.: Small denominators. I. mappings of the circumference onto itself. AMS Trans. Series 2(46), 213–284 (1965)

    Google Scholar 

  8. Jensen, M.H. Bak, P., Bohr, T.: Complete devil’s staircase, fractal dimension and universality of mode-locking structure in the circle map. Phys. Rev. Lett. 50(21), 1637–1639 (1983)

    Article  Google Scholar 

  9. Jensen, M.H. Bak, P., Bohr, T.: Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps. Phys. Rev. A 30(4), 1960–1969 (1984)

    Article  MathSciNet  Google Scholar 

  10. Reichhardt, C., Nori, F.: Phase locking devil’s staircase, Farey trees, and Arnold tongues in driven vortex lattices with periodic pinning. Phys. Rev. Lett. 82(2), 414–417 (1999)

    Article  Google Scholar 

  11. Farey, J.: On a curious property of vulgar functions. J. Philos. Mag. 47, 385–386 (1816)

    Google Scholar 

  12. Wiggins., S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

  13. Ogata, K.: Engenharia de Controle Moderno. Segunda Edição, Prentice Hall do Brasil, São Paulo (1993)

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Authors and Affiliations

  1. Instituto Tecnológico de Aeronautica – ITA, São José dos Campos, Brazil

    Pedro Fernando Almeida Di Donato

  2. Laboratório Associado de Computação e Matemática Aplicada – LAC, Instituto Nacional de Pesquisas Espaciais - INPE, São José dos Campos, Brazil

    Elbert E. N. Macau

  3. Center for Applied Dynamics Research, Kings College, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Celso Grebogi

Authors
  1. Pedro Fernando Almeida Di Donato
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  2. Elbert E. N. Macau
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  3. Celso Grebogi
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Correspondence to Elbert E. N. Macau.

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Donato, P.F.A.D., Macau, E.E.N. & Grebogi, C. Phase locking control in the Circle Map. Nonlinear Dyn 47, 75–82 (2007). https://doi.org/10.1007/s11071-006-9055-7

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  • DOI: https://doi.org/10.1007/s11071-006-9055-7

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