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Communication Dans Un Congrès Année : 2007

Tissular coupling and frequency locking I finite population

Résumé

Synchronization is an extremely important and interesting emergent property of complex systems. The first example found in literature goes back to the 17th century with Christiaan Huygens works. This kind of emergent behavior can be found in artificial systems as well as in natural ones and at many scales (from cell to whole ecological systems). Biology abounds of peri- odic and synchronized phenomena and the works of Ilya Prigogine showed that such behaviors arise within specific conditions: a dissipative structure gener- ally associated to a non-linear dynamic. Biological systems are open, they evolve far from thermodynamic equilibrium and are subject to numerous reg- ulating processes, leading to highly non-linear dynamics. Therefore periodic behaviors appear (with or without synchronization) at any scale. More generally, life itself is governed by circadian rhythms. Those phenomena are as much attractive as they are often spectacular: from cicada populations that appear spontaneously every ten or thirteen years or networks of heart cells that beat together to huge swarms in which fireflies, gathered in a same tree, flash simultaneously. Furthermore, beyond biology one can find a wide source of examples in completely different fields of science (e.g. in behav- ioral psychology with the example of synchronizing applause). For much more artificial and/or theoretical examples, one can consider the whole field of research that studies the coupling of smooth dynamical systems. Nowa- days, it is one of the most important subject related to non-linear systems' dynamics, especially through the notion of chaotic systems' synchronization. This wide source of examples leads the field of research to be highly interdisciplinary, from pure theory to concrete applications and experimen- tations. The classical concept of synchronization is related to the locking of the basic frequencies and instantaneous phases of regular oscillations. Those questions are usually addressed by studying specific kinds of coupled discrete or differential systems, using classical tools of the field. Convinced that synchronization phenomenon is completely natural in a large variety of coupled dynamical systems, we propose a new approach of the subject: firstly, we ask the question of synchronization differently than the usual way. Rather than trying to prove that synchronization actually takes place, we search conditions under which frequencies are locked as soon as the whole system oscillate. Secondly we enlarge the scope of handled models, by building a general framework for coupled systems called " tissular coupling ". This framework is inspired by biological observations at cell's scale, but relevant at any scale of modeling. Under some general assumptions on the kind of interactions that constitute the coupling of the systems, we prove that for a wide class of tissular coupling systems, frequencies are mutually locked to a single value as soon as the whole population is oscillating. This paper exhibits our model of tissular coupling and the frequency locking in the case of a finite number of coupled systems. In the first section we present some mathematical tools and the background we have used in order to study synchronization issue (the results exposed at the end of this paper is only a part of what we have fulfilled, and surely a really small part of what can be done using tissular coupling, this is why we state this framework in its general form). Then, we describe dynamical objects on which we focus, namely the tissular coupling and periodical motions of a population. In the second section we expose a useful way to reduce the problem to a structural one, with no more reference to the dynamics of the coupled systems. In the final section we exhibit some natural conditions under which we are able to prove the main result of this paper, a case of synchronization, in terms of frequencies locking. In a second paper [3] we expose the case of an infinite compact and connected population, which is processed with different mathematical tools.
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Dates et versions

hal-00933203 , version 1 (23-01-2014)

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  • HAL Id : hal-00933203 , version 1

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Laurent Gaubert, Pascal Redou, Jacques Tisseau. Tissular coupling and frequency locking I finite population. EPNACS'07 : Emergent Properties in Natural and Artificial Complex Systems, Oct 2007, Dresden, Germany. pp.121-132. ⟨hal-00933203⟩
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