### Abstract

Recently, there has been a great effort to extract the phase of chaotic attractors and complex oscillators. As a consequence many phases have been introduced, as example the standard phase theta based on the rotation of the vector position, and the phase phi based on the rotation of the tangent vector. Despite of the large interest in the phase dynamics of coupled oscillators there is still a lack of approaches that analyze whether these phase are equivalent and on what conditions these phases work. In this work, we show that the phase phi generalizes the standard phase theta, and it is equal to the length of the Gauss map, the generator of the curvature in differential geometry. Furthermore, we demonstrate, for a broad class of attractors, that the phase synchronization phenomenon between two coherent chaotic oscillators is invariant under the phase definition. Moreover, we discuss to which classes of oscillators the defined phases can be used to calculate quantities as the average frequency and the average period of oscillators. Finally, we generalize the phase phi which allows its use also to homoclinic attractors. (c) 2006 Elsevier B.V. All rights reserved.

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

Pages (from-to) | 159-165 |

Number of pages | 7 |

Journal | Physics Letters A |

Volume | 362 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 26 Feb 2007 |

### Keywords

- phase of chaotic attractors
- coupled oscillators
- neural dynamics
- differential-equations
- synchronization
- model

### Cite this

*Physics Letters A*,

*362*(2-3), 159-165. https://doi.org/10.1016/j.physleta.2006.09.099

**Phase and average period of chaotic oscillators.** / Pereira, T.; Baptista, M. S.; Kurths, J.

Research output: Contribution to journal › Article

*Physics Letters A*, vol. 362, no. 2-3, pp. 159-165. https://doi.org/10.1016/j.physleta.2006.09.099

}

TY - JOUR

T1 - Phase and average period of chaotic oscillators

AU - Pereira, T.

AU - Baptista, M. S.

AU - Kurths, J.

PY - 2007/2/26

Y1 - 2007/2/26

N2 - Recently, there has been a great effort to extract the phase of chaotic attractors and complex oscillators. As a consequence many phases have been introduced, as example the standard phase theta based on the rotation of the vector position, and the phase phi based on the rotation of the tangent vector. Despite of the large interest in the phase dynamics of coupled oscillators there is still a lack of approaches that analyze whether these phase are equivalent and on what conditions these phases work. In this work, we show that the phase phi generalizes the standard phase theta, and it is equal to the length of the Gauss map, the generator of the curvature in differential geometry. Furthermore, we demonstrate, for a broad class of attractors, that the phase synchronization phenomenon between two coherent chaotic oscillators is invariant under the phase definition. Moreover, we discuss to which classes of oscillators the defined phases can be used to calculate quantities as the average frequency and the average period of oscillators. Finally, we generalize the phase phi which allows its use also to homoclinic attractors. (c) 2006 Elsevier B.V. All rights reserved.

AB - Recently, there has been a great effort to extract the phase of chaotic attractors and complex oscillators. As a consequence many phases have been introduced, as example the standard phase theta based on the rotation of the vector position, and the phase phi based on the rotation of the tangent vector. Despite of the large interest in the phase dynamics of coupled oscillators there is still a lack of approaches that analyze whether these phase are equivalent and on what conditions these phases work. In this work, we show that the phase phi generalizes the standard phase theta, and it is equal to the length of the Gauss map, the generator of the curvature in differential geometry. Furthermore, we demonstrate, for a broad class of attractors, that the phase synchronization phenomenon between two coherent chaotic oscillators is invariant under the phase definition. Moreover, we discuss to which classes of oscillators the defined phases can be used to calculate quantities as the average frequency and the average period of oscillators. Finally, we generalize the phase phi which allows its use also to homoclinic attractors. (c) 2006 Elsevier B.V. All rights reserved.

KW - phase of chaotic attractors

KW - coupled oscillators

KW - neural dynamics

KW - differential-equations

KW - synchronization

KW - model

U2 - 10.1016/j.physleta.2006.09.099

DO - 10.1016/j.physleta.2006.09.099

M3 - Article

VL - 362

SP - 159

EP - 165

JO - Physics Letters A

JF - Physics Letters A

SN - 0375-9601

IS - 2-3

ER -