Loss of coherence in dynamical networks: spatial chaos and chimera states

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatio-temporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous R\"ossler systems reveal that intermediate, partially coherent states represent characteristic spatio-temporal patterns at the transition from coherence to incoherence.Comment: 4 pages, 4 figure

An introduction to the synchronization of chaotic systems: coupled skew tent maps

In this tutorial paper, various phenomena linked to the synchronization of chaotic systems are discussed using the simple example of two coupled skew tent maps. The phenomenon of locally riddled basins of attraction is explained using the Lyapunov exponents transversal to the synchronization manifold. The skew tent maps are coupled in two different ways, leading to quite different global dynamic behaviors, especially when ideal system is perturbed by parameter mismatch or noise. The linear coupling leads to intermittent desynchronization bursts of large amplitude, whereas for the nonlinear coupling the synchronization error is asymptotically uniformly bounded

Multistability in the Kuramoto model with synaptic plasticity

We present a simplified phase model for neuronal dynamics with spike timing-dependent plasticity (STDP). For asymmetric, experimentally observed STDP we find multistability: a coexistence of a fully synchronized, a fully desynchronized, and a variety of cluster states in a wide enough range of the parameter space. We show that multistability can occur only for asymmetric STDP, and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies. We test the efficacy of the proposed method on the Kuramoto model which is, de facto, one of the sample models for a description of the phase dynamics in neuronal ensembles

Extreme Sensitivity to Detuning for Globally Coupled Phase Oscillators

Peter Ashwin, Oleksandr Burylko, Yuri Maistrenko, and Oleksandr Popovych, Physical Review Letters, Vol. 96, p. 054102 (2006). "Copyright © 2006 by the American Physical Society."We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N=2, appears at isolated parameter values for N=3 and N=4, and can appear robustly for open sets of parameter values for ≥ 5 oscillators

Mutual synchronization and clustering in randomly coupled chaotic dynamical networks

We introduce and study systems of randomly coupled maps (RCM) where the relevant parameter is the degree of connectivity in the system. Global (almost-) synchronized states are found (equivalent to the synchronization observed in globally coupled maps) until a certain critical threshold for the connectivity is reached. We further show that not only the average connectivity, but also the architecture of the couplings is responsible for the cluster structure observed. We analyse the different phases of the system and use various correlation measures in order to detect ordered non-synchronized states. Finally, it is shown that the system displays a dynamical hierarchical clustering which allows the definition of emerging graphs.Comment: 13 pages, to appear in Phys. Rev.

Riddling : Chimera’s dilemma

We wish to acknowledge the support: Sao Paulo Research Foundation (FAPESP) under Grants 2011/19296-1, 2015/05186-0, 2015/07311-7, 2015/50122-0, and 2017/20920-8, Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico (CNPq), and Coordena¸cao de Aperfei¸coamento de Pessoal de Nıvel Superior (CAPES).Peer reviewedPublisher PD