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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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