We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayed and spatially nonlocal fashion. We find novel clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.
Chimera states, representing a spontaneous break-up of a population of identical oscillators that are identically coupled, into sub-populations displaying synchronized and desynchronized behaviour, have traditionally been found to exist in weakly coupled systems and with some form of nonlocal coupling between the oscillators. Here we show that neither the weak-coupling approximation nor nonlocal coupling are essential conditions for their existence. We obtain for the first time amplitudemediated chimera states in a system of globally coupled complex Ginzburg-Landau oscillators. We delineate the dynamical origins for the formation of such states from a bifurcation analysis of a reduced model equation and also discuss the practical implications of our discovery of this broader class of chimera states.PACS numbers: 05.45. Ra, 05.45.Xt, The spontaneous break-up of a system of identical oscillators, that are identically coupled, into sub-groups of oscillators with different synchronous properties is a fascinating collective phenomenon that was first reported by Kuramoto and Battogtokh[1] and has since been the subject of many investigations . This spatio-temporal pattern of co-existing synchronous and de-synchronous oscillations, named as a chimera state by Abrams and Strogatz [3], has also been experimentally demonstrated in a number of laboratory systems [31][32][33][34][35][36]. The natural manifestation of this state can be seen in such phenomena as unihemispherical sleep in many animals [37,38] where the awake side of the brain shows desynchronized electrical activity, whereas the sleeping side is highly synchronized [8] or in the human brain when in certain regions the neuronal activity gets highly synchronized during epileptic seizures [39] or damage due to Parkinson's disease [40]. In model studies mentioned above chimera states have been found in phase only oscillator systems and in the presence of a nonlocal coupling between the oscillators. This has given rise to a general notion that a weak coupling approximation (implying phase only oscillators) and nonlocal coupling are two essential ingredients for the existence of a chimera state. In a recent work [41] we have demonstrated that the weak coupling approximation is not critical and a more generalized version of the chimera state that includes amplitude effects can be a collective state of the nonlocal complex GinzburgLandau equation (NLCGLE). These amplitude-mediated chimeras (AMCs) can exist as stationary or travelling patterns and show intermittent emergence and decay of amplitude dips in the phase incoherent regions. The next question that naturally arises is whether the nonlocality in the coupling can also be relaxed and whether chimera states can form through other forms of coupling in a system of oscillators. In this paper we address this issue and show for the first time that the amplitude-mediated * e-mail: gautam.sethia@gmail.com chimera state can emerge even in a globally coupled system of oscillators. We discover these states from a numerical...
We investigate the possibility of obtaining chimera state solutions of the non-local Complex Ginzburg-Landau Equation (NLCGLE ) in the strong coupling limit when it is important to retain amplitude variations. Our numerical studies reveal the existence of a variety of amplitude mediated chimera states (including stationary and non-stationary two cluster chimera states), that display intermittent emergence and decay of amplitude dips in their phase incoherent regions. The existence regions of the single-cluster chimera state and both types of two cluster chimera states are mapped numerically in the parameter space of C1 and C2 the linear and nonlinear dispersion coefficients respectively of the NLCGLE. They represent a new domain of dynamical behaviour in the well explored rich phase diagram of this system. The amplitude mediated chimera states may find useful applications in understanding spatio-temporal patterns found in fluid flow experiments and other strongly coupled systems. Chimera states, spatio-temporal patterns of co-existing coherent and incoherent behaviour in an array of coupled identical oscillators, have received a great deal of attention in recent times [1][2][3][4][5]. First found by Kuramoto and Battoghtokh [6] from numerical investigations of the weak coupling version of the non-local Complex Ginzburg-Landau Equation (NLCGLE ), the chimera state has subsequently been studied for a variety of systems [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] including two dimensional ones [24,[27][28][29] and those that have time-delayed coupling [30] or those with a time-delayed feedback [31]. The phase only chimera state has been suggested as a useful paradigm to represent such curious phenomenon as unihemispheric sleep in certain mammals and birds, where during sleep one half of their brain is quiescent while the other half remains active [3,32]. Recently the phase only chimera states have also been observed experimentally in a chemical system [33] , in an opto-electronic set up [34] under controlled laboratory settings as well as in a mechanical experiment consisting of two populations of metronomes [35]. An experimental realization of a modified Ikeda time-delayed equation is also shown to exhibit chimera-like states [36].These past studies have however been confined to the weak coupling limit of the oscillator arrays where the amplitude variations have been ignored and only the dynamical behaviour of the phases have been considered. In many practical situations, such as in fluid flow representations, amplitude equations provide a more realistic description of the physical phenomena and have been widely employed to study the collective behaviour of such systems. An interesting question to ask is therefore whether spatio-temporal patterns corresponding to chimera states can exist for the strong cou- * e-mail: gautam.sethia@gmail.com pling limit. We note here that recently, multi-chimera states have been found in networks of coupled FitzHughNagumo (FHN ) and Hindmarsh-Rose ...
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