Synchronization And Chimera States

Synchronization is a process where two or more individual systems interacting with each other move together in terms of their dynamics. In the 60s of XVII century the longitude problem, i.e., finding a robust, accurate method of the longitude determination for marine navigation was an outstanding challenge. Huygens believed that pendulum clocks, suitably modified to withstand the rigors of the sea, could be sufficiently accurate to reliably determine the longitude. He performed an experiment which showed the tendency of two pendula to synchronize, or anti-synchronize when mounted together on the same beam. Two pendula, mounted together, always ended up swinging in exactly opposite directions, regardless of their respective individual motion. Huygens originally believed that synchronization occured due to air currents shared between two pendula, but after performing several simple tests he dismissed this and attributed the sympathetic motion of pendula to imperceptible movement in the beam from which both pendula were suspended.

 

Other examples where synchronization is observed are fireflies, population of cicada species, pacemaker cells, clapping among audience, voltage in Josephson junction, neutrino oscillations in early universe and a lot more physical and biological systems. There are many types of synchronizations but primarily they can be classified into ’In-phase’ and ’Anti-phase’ synchronizations. Flow on a circle and models like uniform and non-uniform oscillators can be used to study and understand how synchronization arises in systems like fireflies and Josephson junction.

 

In synchronization, the final dynamics usually depend on external parameters and type of coupling. They also depend on similarities of oscillators involved where complete synchrony can be obtained only when all the oscillators are identical but if they are not, they will be phase locked with some steady and stable frequency ratio. In this phenomenon, oscillators always try to catch up with the other ones by either increasing or decreasing their frequencies, thus affecting the current next phase dynamics of the oscillators.

 

One of the famous systems used to model dynamics of a population of oscillators is Kuramoto model. It has sinusoidal type of coupling in its oscillators in a given network.The model makes certain basic assumptions like weak coupling and oscillators being identical or nearly identical.

 

There is an interesting phenomenon observed in synchronization called ‘Chimera states‘. Chimera or also known as spatio-temporal dynamics is a phenomenon having coexistence of both synchrony and asynchrony in a given population of oscillators coupled to each other. It can also be seen as spatial coexistence of coherent and incoherent oscillators. Unlike partial asynchrony observed in a system where synchronization is taking place, these chimera states are stable.

 

Until Kuramoto and his colleagues made observations on synchronization while trying to simulate the dynamics of Ginzburg-Landau equation(a modulation that describes the evolution of small perturbations of a marginally unstable basic state of a system of nonlinear partial differential equations on an unbounded domain), mathematicians and physicists thought synchrony and disorder are mutually exclusive for a given system. Later on, many people including Strogatz found that both synchrony and disorder can arise simultaneously in a given population of identical oscillators given that the coupling is non local(the coupling strength decreases with respect to the distance between oscillators). It was also observed that no fine tuning of initial conditions or specific cases are required to get chimera behaviour.

 

Chimera states were observed in neuronal models, chaotic systems, complex networks, time varying networks, modular types and Hopf normal forms. Recently, they have been observed numerically in systems with topology of global, local and one way coupling too. Experimental evidence of these have been observed in optical coupled maps, coupled chemical oscillators, metronomes and squid meta-materials. Application of chimera can be seen in brain diseases like Parkinson’s, epileptic seizures, Alzheimer’s, brain tumors and also in slow wave sleep of some aquatic animals and migrated birds where half part of the brain is active and the other half is in sleep.

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