Chimeras and Clusters Emerging from Robust‐Chaos Dynamics

Abstract

We show that dynamical clustering, where a system segregates into distinguishable subsets of synchronized elements, and chimera states, where differentiated subsets of synchronized and desynchronized elements coexist, can emerge in networks of globally coupled robust-chaos oscillators. We describe the collective behavior of a model of globally coupled robust-chaos maps in terms of statistical quantities and characterize clusters, chimera states, synchronization, and incoherence on the space of parameters of the system. We employ the analogy between the local dynamics of a system of globally coupled maps with the response dynamics of a single driven map. We interpret the occurrence of clusters and chimeras in a globally coupled system of robust-chaos maps in terms of windows of periodicity and multistability induced by a drive on the local robust-chaos map. Our results show that robust-chaos dynamics does not limit the formation of cluster and chimera states in networks of coupled systems, as it had been previously conjectured.

Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps

“…Also, formula ( 5) is quite suitable for numerical experiments, since the principal numerical cost to obtain the TLE at some point comes from the computation of the orbit of this point. Chimeras have been characterized and detected by means of the large times behavior of other indicators reflecting the proportion of units that are synchronized with at least one other unit [7,8,10,21]. For each orbit {x t } t∈N and each time steps, these indicators need to compute the distance |x t i − x t j | between the state of each units [7,8,10] and then the distribution of this quantity [21].…”

“…Chimeras have been characterized and detected by means of the large times behavior of other indicators reflecting the proportion of units that are synchronized with at least one other unit [7,8,10,21]. For each orbit {x t } t∈N and each time steps, these indicators need to compute the distance |x t i − x t j | between the state of each units [7,8,10] and then the distribution of this quantity [21]. Since they involve all the information on the distance between the states of each unit, they are powerful indicators to distinguish different dynamical patterns, but require more numerical resources (of the order of N 2 at each time step) than the TLE.…”

“…While chimera states in networks of oscillators were initially associated with non-local coupling and the breaking of symmetry resulting from such coupling [1,26], numerical evidence has suggested their emergence even in systems with significant symmetries in the coupling structure. For instance, in the recent years, chimeras states have been detected in globally coupled complex Ginzburg-Landau equations [28] and in discrete time systems such as globally coupled maps (GCM) for different kind of maps including those with robust chaotic behavior [4,7,8,10,19,20]. Overall, the conditions, on the coupling structure and the individual dynamics of the coupled units, allowing the appearance of attracting chimera states has been an important subject since their discovery.…”

“…Usually, chimeras are numerically detected by means of indicators involving the computation of the distance between the states of all the coupled units, in order to estimate the proportion of synchronized units [7,8,10,21]. On the one hand it allows accurate classifications of the encountered chimera states, but on the other hand it may require important numerical resources.…”