Emergence of multicluster chimera states

We thank Prof. L. Huang for helpful discussions. This work was partially supported by ARO under Grant No. W911NF-14-1-0504 and by NSF of China under Grant No. 11275003. The visit of NY to Arizona State University was partially sponsored by Prof. Z. Zheng and the State Scholarship Fund of China.Peer reviewedPublisher PD

Emergence of multicluster chimera states

abstract: A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.The final version of this article, as published in Scientific Reports, can be viewed online at: https://www.nature.com/articles/srep1298

Transition from chimera/solitary states to traveling waves

We study numerically the spatiotemporal dynamics of a ring network of nonlocally coupled nonlinear oscillators, each represented by a two-dimensional discrete-time model of the classical van der Pol oscillator. It is shown that the discretized oscillator exhibits a richer behavior, combining the peculiarities of both the original system and its own dynamics. Moreover, a large variety of spatiotemporal structures is observed in the network of discrete van der Pol oscillators when the discretization parameter and the coupling strength are varied. Such regimes as the coexistence of multichimera state/traveling wave and solitary state are revealed for the first time and studied in detail. It is established that the majority of the observed chimera/solitary states, including the newly found ones, are transient towards the purely traveling wave mode. The peculiarities of the transition process and the lifetime (transient duration) of the chimera structures and the solitary state are analyzed depending on the system parameters, observation time, initial conditions, and influence of external noise

Synchronized clusters in coupled map networks. II. Stability analysis

We study self-organized and driven synchronization in some simple coupled map networks, namely globally coupled networks and complete bipartite networks, using both linear stability analysis and Lyapunov function approach and determine stability conditions for synchronization. The phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in Part I. Lyapunov function approach shows that when any two nodes are in driven synchronization, all the coupling terms in the difference between the variables of these two nodes cancel out, whereas when they are in self-organized synchronization, the direct coupling term between the two nodes adds an extra term while the other couplings cancel out. We also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional Lyapunov exponent about zero can be a criterion for its occurrence

Hierarchical frequency clusters in adaptive networks of phase oscillators

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 29, 103134 (2019) and may be found at https://doi.org/10.1063/1.5097835.Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency multiclusters by investigating the effect of the time scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multiclusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.DFG, 411803875, Dynamik gekoppelter Systeme mit Zeitverzögerungen und deren AnwendungenDFG, 308748074, Komplexe dynamische Netzwerke: Effekte von heterogenen, adaptiven und zeitverzögerten Kopplunge

Adaptive dynamical networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks

Filtering Suppresses Amplitude Chimeras

Amplitude chimera (AC) is an interesting chimera pattern that has been discovered recently and is distinct from other chimera patterns, like phase chimeras and amplitude mediated phase chimeras. Unlike other chimeras, in the AC pattern all the oscillators have the same phase velocity, however, the oscillators in the incoherent domain show periodic oscillations with randomly shifted origin. In this paper we investigate the effect of local filtering in the coupling path on the occurrence of AC patterns. Our study is motivated by the fact that in the practical coupling channels filtering effects come into play due to the presence of dispersion and dissipation. We show that a low-pass or all-pass filtering is actually detrimental to the occurrence of AC. We quantitatively establish that with decreasing cut-off frequency of the filter, an AC transforms into a synchronized pattern. We also show that the symmetry-breaking steady state, i.e., the oscillation death state can be revoked and rhythmogenesis can be induced by local filtering. Our study will shed light on the understanding of many biological systems where spontaneous symmetry-breaking and local filtering occur simultaneously.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept