Introduction to focus issue: Synchronization in large networks and continuous media - Data, models, and supermodels

Abstract

The synchronization of loosely coupled chaotic systems has increasingly found applications to large networks of differential equations and to models of continuous media. These applications are at the core of the present Focus Issue. Synchronization between a system and its model, based on limited observations, gives a new perspective on data assimilation. Synchronization among different models of the same system defines a supermodel that can achieve partial consensus among models that otherwise disagree in several respects. Finally, novel methods of time series analysis permit a better description of synchronization in a system that is only observed partially and for a relatively short time. This Focus Issue discusses synchronization in extended systems or in components thereof, with particular attention to data assimilation, supermodeling, and their applications to various areas, from climate modeling to macroeconomics. Synchronization among regular oscillators such as a complex organism's circadian rhythms, pendulum clocks on a common wall, or blinking fireflies establishes a surprising order in natural systems. Theoretical and numerical results obtained over the past 25 years with coupled chaotic systems suggest that synchronistic relationships could possibly occur between systems whose internal behavior is not ostensibly regular, extending greatly the potential range of synchronism in nature. More recently, synchronization has been explored in naturally occurring, chaotic systems with very large numbers of variables and in models thereof; the latter are typically governed by sets of ordinary differential equations (ODEs) on large networks or by partial differential equations (PDEs) on continuous media. An important instance is the synchronization between a system and its real-time computational model that can be induced by a limited set of observations of the system. Such truth–model synchronization corresponds to the well-established practice of data assimilation, used extensively in meteorology, oceanography, and the climate sciences in general. An extension of this idea is to allow a set of alternative models of the same real system to synchronize with one another, as well as the real system, by exchanging data and thus forming a supermodel. Such a supermodel offers a potential solution to problems of divergent predictions by different expert models, and it has been shown to improve upon the common practice of merely averaging over model outputs. This Focus Issue sheds further light on the uses of synchronization for data assimilation and for supermodeling, as well as on current developments in synchronization within and between extended systems, natural and social, in general.