Chimeralike states in networks of bistable time-delayed feedback oscillators coupled via the mean field

Quasinormal Forms for Chains of Coupled Logistic Equations with Delay

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

Response of solitary states to noise-modulated parameters in nonlocally coupled networks of Lozi maps

We study numerically the impact of heterogeneity in parameters on the dynamics of nonlocally coupled discrete-time systems, which exhibit solitary states along the transition from coherence to incoherence. These partial synchronization patterns are described as states when single or several elements demonstrate different dynamics compared with the behavior of other elements in a network. Using as an example a ring network of nonlocally coupled Lozi maps, we explore the robustness of solitary states to heterogeneity in parameters of local dynamics or coupling strength. It is found that if these network parameters are continuously modulated by noise, solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence are constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient.

Control of inter-layer synchronization by multiplexing noise

We study the synchronization of spatio-temporal patterns in a two-layer network of coupled chaotic maps, where each layer is represented by a nonlocally coupled ring. In particular, we focus on noisy inter-layer communication that we call multiplexing noise. We show that noisy modulation of inter-layer coupling strength has a significant impact on the dynamics of the network and specifically on the degree of synchronization of spatio-temporal patterns of interacting layers initially (in the absence of interaction) exhibiting chimera states. Our goal is to develop control strategies based on multiplexing noise for both identical and non-identical layers. We find that for the appropriate choice of intensity and frequency characteristics of parametric noise, complete or partial synchronization of the layers can be observed. Interestingly, for achieving inter-layer synchronization through multiplexing noise, it is crucial to have colored noise with intermediate spectral width. In the limit of white noise, the synchronization is destroyed. These results are the first step toward understanding the role of noisy inter-layer communication for the dynamics of multilayer networks.

Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps

We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.