Tissular coupling and frequency locking II infinite population

Laurent Gaubert (1) , Pascal Redou (1) , Jacques Tisseau (1)

1 CERV - Centre Européen de Réalité Virtuelle

In an accompanying paper, we addressed the issue of synchronization under the framework of tissular coupling and in the case of a finite population. This issue is one of the main dynamical emergent property of complex systems. Indeed, this phenomenon actually occurs in natural systems at many scales (from cell to whole ecological systems). Many examples can be found in various fields of experimentation and theoretical research. It is in particular a key concept in the field of non- linear systems' dynamics, especially through the notion of chaotic systems' synchronization. This wide source of examples explain why this field of research is highly interdisciplinary, from pure theory to concrete applications and experimentations. The classical concept of synchronization is related to the locking of the basic frequencies and instantaneous phases of regular oscillations. Those issues are usually addressed by studying specific coupled systems, using classical tools of the field. Convinced that synchronization phenomenon is completely natural in a large variety of coupled dynamical systems, we proposed in a preceding paper a new approach to the subject: we built a general model of coupled systems, called " tissular coupling ". Within natural assumptions on interactions of those coupled systems, we proved that, for a wide class of autonomous differential systems, as soon as a finite population exhibits oscillating behaviors, their frequencies are mutually locked to a single value. This paper presents a new piece of answer to those synchronization issues as it addresses the issue of an infinite compact connected population. Even if the general questions addressed in this paper are similar to those addressed in the later one, this study is done with completely different techniques than in the finite situation and the results achieved require assumptions that are specific to the continuous situation. This paper stresses the fact that synchronization is a very natural phenomenon that can be find in many kind of coupled dynamical systems. In the first section, we recall some parts of the framework already introduced and some mathematical tools specific to our problem. Then we address a sub problem which deals with diffuse coupling, concluding with a general result on synchronization. Finally, we go back to the general case of a uniformly linear tissular coupling and show how those dynamical systems, and based on which assumptions, exhibits the frequencies locking property

Domaines: Systèmes dynamiques [math.DS]

Laurent Gaubert, Pascal Redou, Jacques Tisseau. Tissular coupling and frequency locking II infinite population. EPNACS'07 : Emergent Properties in Natural and Artificial Complex Systems, Oct 2007, Dresden, Germany. pp.133-140. ⟨hal-00933209⟩