Chimera states: Effects of different coupling topologies

Collective behavior among coupled dynamical units can emerge in various forms as a result of different coupling topologies as well as different types of coupling functions. Chimera states have recently received ample attention as a fascinating manifestation of collective behavior, in particular describing a symmetry breaking spatiotemporal pattern where synchronized and desynchronized states coexist in a network of coupled oscillators. In this perspective, we review the emergence of different chimera states, focusing on the effects of different coupling topologies that describe the interaction network connecting the oscillators. We cover chimera states that emerge in local, nonlocal and global coupling topologies, as well as in modular, temporal and multilayer networks. We also provide an outline of challenges and directions for future research.Comment: 7 two-column pages, 4 figures; Perspective accepted for publication in EP

Rapid convergence of time-averaged frequency in phase synchronized systems

Numerical and experimental evidence is presented to show that many phase synchronized systems of non-identical chaotic oscillators, where the chaotic state is reached through a period-doubling cascade, show rapid convergence of the time-averaged frequency. The speed of convergence toward the natural frequency scales as the inverse of the measurement period. The results also suggest an explanation for why such chaotic oscillators can be phase synchronized.Comment: 6 pages, 9 figure

Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation

We present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at Hopf bifurcation. For a simple model consisting of just two oscillators with a time delayed coupling, the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death, phase locked and incoherent regions. A novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators. Similar results are obtained for an array of N oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified. Some general analytic results for the N tending to infinity (thermodynamic) limit are also obtained and the implications of the time delay effects for physical applications are discussed.Comment: 20 aps formatted revtex pages (including 13 PS figures); Minor changes over the previous version; To be published in Physica

Intermittent chaotic chimeras for coupled rotators

Two symmetrically coupled populations of N oscillators with inertia $m$ display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendula. In particular, we report the first evidence of intermittent chaotic chimeras, where one population is synchronized and the other jumps erratically between laminar and turbulent phases. These states have finite life-times diverging as a power-law with N and m. Lyapunov analyses reveal chaotic properties in quantitative agreement with theoretical predictions for globally coupled dissipative systems.Comment: 6 pages, 5 figures SUbmitted to Physical Review E, as Rapid Communicatio

Characterizing correlations and synchronization in collective dynamics

Synchronization, that occurs both for non-chaotic and chaotic systems, is a striking phenomenon with many practical implications in natural phenomena. However, even before synchronization, strong correlations occur in the collective dynamics of complex systems. To characterize their nature is essential for the understanding of phenomena in physical and social sciences. The emergence of strong correlations before synchronization is illustrated in a few piecewise linear models. They are shown to be associated to the behavior of ergodic parameters which may be exactly computed in some models. The models are also used as a testing ground to find general methods to characterize and parametrize the correlated nature of collective dynamics.Comment: 37 pages, 37 figures, Late

Coherent periodic activity in excitatory Erdos-Renyi neural networks:The role of network connectivity

We consider an excitatory random network of leaky integrate-and-fire pulse coupled neurons. The neurons are connected as in a directed Erd\"os-Renyi graph with average connectivity $$ scaling as a power law with the number of neurons in the network. The scaling is controlled by a parameter $\gamma$, which allows to pass from massively connected to sparse networks and therefore to modify the topology of the system. At a macroscopic level we observe two distinct dynamical phases: an Asynchronous State (AS) corresponding to a desynchronized dynamics of the neurons and a Partial Synchronization (PS) regime associated with a coherent periodic activity of the network. At low connectivity the system is in an AS, while PS emerges above a certain critical average connectivity $_c$. For sufficiently large networks, $_c$ saturates to a constant value suggesting that a minimal average connectivity is sufficient to observe coherent activity in systems of any size irrespectively of the kind of considered network: sparse or massively connected. However, this value depends on the nature of the synapses: reliable or unreliable. For unreliable synapses the critical value required to observe the onset of macroscopic behaviors is noticeably smaller than for reliable synaptic transmission. Due to the disorder present in the system, for finite number of neurons we have inhomogeneities in the neuronal behaviors, inducing a weak form of chaos, which vanishes in the thermodynamic limit. In such a limit the disordered systems exhibit regular (non chaotic) dynamics and their properties correspond to that of a homogeneous fully connected network for any $\gamma$-value. Apart for the peculiar exception of sparse networks, which remain intrinsically inhomogeneous at any system size.Comment: 7 pages, 11 figures, submitted to Chao