### 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.

Original language | English |
---|---|

Pages (from-to) | 75-82 |

Number of pages | 8 |

Journal | Nonlinear Dynamics |

Volume | 47 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jan 2007 |

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### Keywords

- phase-locking
- Farey Series
- Circle Map

### Cite this

*Nonlinear Dynamics*,

*47*(1-3), 75-82. https://doi.org/10.1007/s11071-006-9055-7

**Phase locking control in the Circle Map.** / Almeida Di Donato, Pedro Fernando; Macau, Elbert E. N.; Grebogi, Celso.

Research output: Contribution to journal › Article

*Nonlinear Dynamics*, vol. 47, no. 1-3, pp. 75-82. https://doi.org/10.1007/s11071-006-9055-7

}

TY - JOUR

T1 - Phase locking control in the Circle Map

AU - Almeida Di Donato, Pedro Fernando

AU - Macau, Elbert E. N.

AU - Grebogi, Celso

PY - 2007/1

Y1 - 2007/1

N2 - 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.

AB - 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.

KW - phase-locking

KW - Farey Series

KW - Circle Map

U2 - 10.1007/s11071-006-9055-7

DO - 10.1007/s11071-006-9055-7

M3 - Article

VL - 47

SP - 75

EP - 82

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 1-3

ER -