Today's strongly connected, global networks have produced highly interdependent systems that we do not understand and cannot control well. These systems are vulnerable to failure at all scales, posing serious threats to society, even when external shocks are absent. As the complexity and interaction strengths in our networked world increase, man-made systems can become unstable, creating uncontrollable situations even when decision-makers are well-skilled, have all data and technology at their disposal, and do their best. To make these systems manageable, a fundamental redesign is needed. A 'Global Systems Science' might create the required knowledge and paradigm shift in thinking.

Globally networked risks and how to respond

Chimera states describing the stable coexistence of synchronous and incoherent dynamics have so far only been realized numerically. An experimental demonstration of these states in a network of discrete chemical oscillators reveals behaviour that differs from that predicted by existing phase-oscillator models.

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Chimera and phase-cluster states in populations of coupled chemical oscillators

A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.

http://dmabrams.esam.northwestern.edu/pubs/Panaggio%20and%20Abrams%20-%20Chimera%20states%20(review)%20-%20Nonlinearity%2028,%20R67-R87%20(2015).pdf

Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in "flocks" of oscillators or dynamic elements.

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On partial contraction analysis for coupled nonlinear oscillators

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.

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Solvable model for chimera states of coupled oscillators

Cycling attractors of coupled cell systems and dynamics with symmetry- Modelling diversity by chaos and classification by synchronization- Basic Principles of Direct Chaotic Communications- Prevalence of Milnor Attractors and Chaotic Itinerancy in 'High'-dimensional Dynamical Systems- Generalization of the Feigenbaum-Kadanoff-Shenker Renormalization and Critical Phenomena Associated with the Golden Mean Quasiperiodicity- Synchronization and clustering in ensembles of coupled chaotic oscillators- Nonlinear Phenomena in Nephron-Nephron Interaction- Synchrony in Globally Coupled Chaotic, Periodic, and Mixed Ensembles of Dynamical Units- Phase synchronization of regular and chaotic self-sustained oscillators- Control of dynamical systems via time-delayed feedback and unstable controller

I︠u︡. L. Maĭstrenko

Papers

Synchronization: Theory and Application