Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 436198, 10 pages
doi:10.1155/2011/436198
Research Article

Specific Mathematical Aspects of Dynamics Generated by Coherence Functions

1Department of Electrical and Computer Engineering, University of West Florida, 11000 University Parkway, Pensacola, Fl 32514, USA
2Faculty of Applied Sciences, Politechnica University, Hagi-Ghita 81, 060032 Bucharest, Romania

Received 18 October 2010; Accepted 27 October 2010

Academic Editor: Ming Li

Copyright © 2011 Ezzat G. Bakhoum and Cristian Toma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 integrodifferential 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.