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**Specific mathematical aspects of dynamics generated by coherence functions.**
*(English)*
Zbl 1248.37075

Summary: This study presents specific aspects of dynamics generated by the coherence function (acting in an integral manner). It is considered that an oscillating system starting to work from initial nonzero conditions is commanded by the coherence function between the output of the system and an alternating function of a certain frequency. For different initial conditions, the evolution of the system is analyzed. The equivalence between integro-differential equations and integral equations implying the same number of state variables is investigated; it is shown that integro-differential equations of second order are far more restrictive regarding the initial conditions for the state variables. Then, the analysis is extended to equations of evolution where the coherence function is acting under the form of a multiple integral. It is shown that for the coherence function represented under the form of an \(n\)th integral, some specific aspects as multiscale behaviour suitable for modelling transitions in complex systems (e.g., quantum physics) could be noticed when \(n\) equals \(4\), \(5\), or \(6\).

### MSC:

37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |

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\textit{E. G. Bakhoum} and \textit{C. Toma}, Math. Probl. Eng. 2011, Article ID 436198, 10 p. (2011; Zbl 1248.37075)

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### References:

[1] | M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002 |

[2] | News from National Institute of Standards and Technology (NIST), “Quantum Cats Made of Light,” 2010, to appear, http://www.photonics.com/Article.aspx?AID=44194. |

[3] | T. Gerrits, S. Glancy, T. Clement, et al., “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Physical Review A, vol. 82, no. 3, 4 pages, 2010. |

[4] | B. Lazar, A. Sterian, S. t. Pusca, V. Paun, C. Toma,, and C. Morarescu, “Simulating delayed pulses in organic materials,” Lecture Notes Computer Science, vol. 3980, pp. 779-785, 2006. |

[5] | G. Toma and F. Doboga, “Vanishing waves on closed intervals and propagating short-range phenomena,” Mathematical Problems in Engineering, vol. 2008, Article ID 359481, 14 pages, 2008. · Zbl 1177.35022 |

[6] | E. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 13 pages, 2010. · Zbl 1191.35219 |

[7] | V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations With Periodic Coefficients, Krieger Publishing Company, Malabar, Fla, USA, 1975. · Zbl 0308.34001 |

[8] | J. Puig, Reducibility of Linear Differential Equations with Quasi-Periodic Coefficients: A Survey, University of Barcelona, Barcelona, Spain, 2002. |

[9] | M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368-1372, 2010. |

[10] | M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008. |

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