### Abstract

We study the dynamics and synchronization properties of a system of complex non-linear equations describing detuned lasers. These equations possess a whole circle of fixed points, while the corresponding real variable equations have only isolated fixed points. We examine the stability of their equilibrium points and determine conditions under which the complex equations have positive, negative or zero Lyapunov exponents and chaotic, quasiperiodic or periodic attractors for a wide range of parameter values. We investigate the synchronization of chaotic solutions of our detuned laser system, using as a drive a similar set of equations and applying the method of global synchronization. We find attractors whose three-dimensional projection is not at all similar to the well-known shape of the (real) Lorenz attractor. Finally, we apply complex periodic driving to the electric field equation and show that the model can exhibit a transition from chaotic to quasiperiodic oscillations. This leads us to the discovery of an exact periodic solution, whose amplitude and frequency depend on the parameters of the system. Since this solution is stable for a wide range of parameter values, it may be used to control the system by entraining it with the applied periodic forcing.

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

Pages (from-to) | 63-79 |

Number of pages | 17 |

Journal | Dynamical Systems |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2009 |

Externally published | Yes |

### Fingerprint

### Keywords

- Attractors
- Chaos
- Control
- Detuned laser systems
- Periodic forcing
- Synchronization

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications

### Cite this

*Dynamical Systems*,

*24*(1), 63-79. https://doi.org/10.1080/14689360802438298

**Dynamical properties and synchronization of complex non-linear equations for detuned lasers.** / M. Mahmoud, Gamal; Bountis, T.; Al-Kashif, M. A.; Aly, Shaban A.

Research output: Contribution to journal › Article

*Dynamical Systems*, vol. 24, no. 1, pp. 63-79. https://doi.org/10.1080/14689360802438298

}

TY - JOUR

T1 - Dynamical properties and synchronization of complex non-linear equations for detuned lasers

AU - M. Mahmoud, Gamal

AU - Bountis, T.

AU - Al-Kashif, M. A.

AU - Aly, Shaban A.

PY - 2009/3

Y1 - 2009/3

N2 - We study the dynamics and synchronization properties of a system of complex non-linear equations describing detuned lasers. These equations possess a whole circle of fixed points, while the corresponding real variable equations have only isolated fixed points. We examine the stability of their equilibrium points and determine conditions under which the complex equations have positive, negative or zero Lyapunov exponents and chaotic, quasiperiodic or periodic attractors for a wide range of parameter values. We investigate the synchronization of chaotic solutions of our detuned laser system, using as a drive a similar set of equations and applying the method of global synchronization. We find attractors whose three-dimensional projection is not at all similar to the well-known shape of the (real) Lorenz attractor. Finally, we apply complex periodic driving to the electric field equation and show that the model can exhibit a transition from chaotic to quasiperiodic oscillations. This leads us to the discovery of an exact periodic solution, whose amplitude and frequency depend on the parameters of the system. Since this solution is stable for a wide range of parameter values, it may be used to control the system by entraining it with the applied periodic forcing.

AB - We study the dynamics and synchronization properties of a system of complex non-linear equations describing detuned lasers. These equations possess a whole circle of fixed points, while the corresponding real variable equations have only isolated fixed points. We examine the stability of their equilibrium points and determine conditions under which the complex equations have positive, negative or zero Lyapunov exponents and chaotic, quasiperiodic or periodic attractors for a wide range of parameter values. We investigate the synchronization of chaotic solutions of our detuned laser system, using as a drive a similar set of equations and applying the method of global synchronization. We find attractors whose three-dimensional projection is not at all similar to the well-known shape of the (real) Lorenz attractor. Finally, we apply complex periodic driving to the electric field equation and show that the model can exhibit a transition from chaotic to quasiperiodic oscillations. This leads us to the discovery of an exact periodic solution, whose amplitude and frequency depend on the parameters of the system. Since this solution is stable for a wide range of parameter values, it may be used to control the system by entraining it with the applied periodic forcing.

KW - Attractors

KW - Chaos

KW - Control

KW - Detuned laser systems

KW - Periodic forcing

KW - Synchronization

UR - http://www.scopus.com/inward/record.url?scp=61449172878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=61449172878&partnerID=8YFLogxK

U2 - 10.1080/14689360802438298

DO - 10.1080/14689360802438298

M3 - Article

VL - 24

SP - 63

EP - 79

JO - Dynamical Systems

JF - Dynamical Systems

SN - 1468-9367

IS - 1

ER -