Abstract

Synchronization, which occurs for both chaotic and nonchaotic systems, is a striking phenomenon with many practical implications for natural phenomena and technological applications. However, even before synchronization, strong correlations and complex patterns occur in the collective dynamics of natural systems. To characterize their nature is essential for understanding many phenomena in physical and social sciences as well as the perspectives to control their behavior. Because simple correlation measures are unable to characterize these collective patterns, we have developed more general methods for their detection and parametrization. The emergence of patterns of strong correlations before synchronization is illustrated in a few models. They are shown to be associated with the behavior of ergodic parameters. The models are then used as a testing ground of the new pattern characterization tools.

Highlights

  • When natural systems are made to interact with one another, collective properties emerge that would be hard to predict from their individual properties

  • Synchronization is not the whole story because, without or before synchronization, much subtler correlations occur in the global dynamical behavior of interacting systems

  • How the techniques discussed do provide information on the correlations of the collective dynamics will be clear by their application to a few models

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Summary

Introduction

When natural systems are made to interact with one another, collective properties emerge that would be hard to predict from their individual properties. The central question here concerns the emergence of coherent behavior: synchronization or other types of correlation This occurs for systems with regular behavior as well as for systems that have chaotic dynamics (lasers, neural networks, physiological processes, etc.). Mostly numerical, has been done on partially synchronized states, clustering, dimensional reduction and so on [47,48,49,50,51,52,53,54] Striking as it is, synchronization is not the whole story because, without or before synchronization, much subtler correlations occur in the global dynamical behavior of interacting systems. Most of the numerical illustrations of the results are presented for a small (~100) number of interacting agents; in all cases, to exclude finite size effects, simulations with much larger numbers were performed with qualitatively similar results

The Deformed Kuramoto Model
Characterizing Correlations
Illustration of the Tools on Some Models
Dynamical Clustering
The Conditional Exponents Spectrum
Conclusion
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