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Edited by: Jun Ma, Lanzhou University of Technology, China

Reviewed by: Tanmoy Banerjee, University of Burdwan, India; Carlo Laing, College of Sciences, Massey University, New Zealand; V. K. Chandrasekar, SASTRA University, India

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Time delay in complex networks with multiple interacting layers gives rise to special dynamics. We study the scenarios of time delay induced patterns in a three-layer network of FitzHugh-Nagumo oscillators. The topology of each layer is given by a nonlocally coupled ring. For appropriate values of the time delay in the couplings between the nodes, we find chimera states, i.e., hybrid spatio-temporal patterns characterized by coexisting domains with incoherent and coherent dynamics. In particular, we focus on the interplay of time delay in the intra-layer and inter-layer coupling term. In the parameter plane of the two delay times we find regions where chimera states are observed alternating with coherent dynamics. Moreover, in the presence of time delay we detect full and relay inter-layer synchronization.

During the early eighteenth century, Leonhard Euler published a paper on The Seven Bridges of Königsberg providing a mathematical background on vertices and edges [

Recently, various synchronization scenarios have been investigated in multilayer structures, including remote and relay synchronization [

Chimera state is a peculiar partial synchronization pattern that refers to a hybrid dynamics where coherence and incoherence emerge simultaneously in a network of identical oscillators [

We study a multiplex network with three layers (triplex) as shown in

where _{i} and the inter-layer delay time is τ_{ij}. The coupling radius in layer _{i}. The local dynamics of each oscillator is given by

where ε = 0.05 is the parameter characterizing the time scale separation. The FHN oscillator exhibits either oscillatory (|_{i} stands for the coupling strength inside the layer (intra-layer coupling), and σ_{ij} is the inter-layer coupling. For an ordinal inter-layer coupling with constant row sum we set σ_{12} = σ_{23}, which yields the inter-layer coupling matrix

Triplex network with ordinal coupling: The middle layer _{i} and time delay τ_{i}, and the inter-layer coupling is characterized by the strength σ_{ij} and time delay τ_{ij}. For example, in layer 3 the intra-layer coupling strength is given by σ_{3} and the intra-layer time delay is τ_{3}. Similarly, for the layers 1 and 2 the inter-layer coupling strength is given by σ_{12} and the inter-layer time delay is τ_{12}.

The connections between the nodes are given by the diffusive coupling with the following coupling matrix

and coupling phase

Chimera states are spatio-temporal patterns where incoherent and coherent domains coexist in space. For certain values of coupling strength σ_{i}_{i}

To provide an overview of the patterns observed in the network, we calculate the map of regimes in the parameter plane of intra-layer delay time τ_{i}_{ij}_{k} = 2π_{k}/Δ_{k} denotes the number of complete rotations performed by the _{k}, on the other hand, by means of the Laplacian distance measure [_{trans} = 2, 000. All simulations are evaluated then after Δ

Dynamical regimes in the parameter plane of intra-layer coupling delay τ_{i} ≡ τ_{1} = τ_{2} = τ_{3} and inter-layer coupling delay τ_{ij} ≡ τ_{12} = τ_{23}: “salt & pepper” states (green islands) occur in the region of coherent states (blue region) as traveling waves, cluster or synchronized states. At the border between these two regimes chimera states can be found (red color). We distinguish between the different regions on the one hand, by analyzing the mean phase velocity and a snapshot of variables _{k}, on the other hand, by means of the Laplacian distance measure [_{i}, τ_{ij}) plane has been sampled in steps Δτ_{i} = 0.05 and Δτ_{ij} = 0.1. For all simulations of Equation (1) random initial conditions are taken. Parameters are chosen as ε = 0.05, _{i} = 0.2, σ_{ij} = 0.05, _{i} = 170,

Dynamics in all three layers for different values of the inter-layer delay time τ_{ij}: _{ij} = 0.4, _{ij} = 0.9, _{ij} = 1.7. The intra-layer delay time is fixed at τ_{i} = 0.8. The left column displays snapshots of variables _{k} (dark blue) for the individual layers and the local inter-layer synchronization error

Dynamics in all three layers for different values of the intra-layer delay time τ_{i}: _{i} = 2.8, _{i} = 2.6, _{i} = 2.4. The inter-layer delay time is fixed at τ_{ij} = 2.6. The left column displays snapshots of variables _{k} (dark blue) for the individual layers and the local inter-layer synchronization error

In many systems with time delays resonance effects can be expected if the delay time is an integer or half-integer multiple of the period in the uncoupled system [_{ij} we can observe this effect for half-integer multiples of the period _{i} ≈ 2.5). Concerning the intra-layer delay time τ_{i} we find a resonance effect in the case of integer multiple of the delay. For greater values of the delay times τ_{i} and τ_{ij} the dynamical regions are becoming curved (see green islands in _{i} ≈ 2.5). This can be explained by the fact that branches of periodic solutions, which are reappearing for integer multiples of the intrinsic period, are becoming stretched with increasing delay time [_{i} ≈ 0.5, the green islands at τ_{i} ≈ 2.5 are rotated clockwise by approximate π/8. The consequence is an overlapping of the delay islands for small intra-layer delay time τ_{i}, whereas for larger delays the islands become separated.

Networks with multiple layers demonstrate remote synchronization of distant layers via a relay layer. Regarding the inter-layer synchronization, two synchronization mechanisms are conceivable in a triplex network:

full inter-layer synchronization when synchronization is observed between all the layers and

relay inter-layer synchronization when synchronization occurs exclusively between the two outer layers.

A useful measure for synchronization between two layers ^{ij} [

where ∥·∥ denotes the Euclidean norm. One can distinguish between the two synchronization mechanisms by measuring the global inter-layer synchronization error between the first and second layer ^{12} and between the first and third layer ^{13}: In the case of full inter-layer synchronization ^{12} = 0 as well as ^{13} = 0, while in the case of relay inter-layer synchronization ^{12} ≠ 0 and ^{13} = 0.

To provide more insight into the synchronizability of patterns between two layers

The local inter-layer synchronization error is convenient for detecting the synchronized nodes between two layers. In _{i} and τ_{ij} we can find full inter-layer synchronization (see

In conclusion, we have studied chimera states in a three-layer network of FitzHugh-Nagumo oscillators, where each layer has a nonlocal coupling topology. Focusing on the role of time delays in the coupling terms and their influence on chimera states, we have performed a numerical study of complex spatio-temporal patterns in the network. In the parameter plane of the intra-layer τ_{i} and the inter-layer τ_{ij} time delay, we have determined the regions where chimera patterns occur, alternating with regimes of coherent states. A proper choice of time delay allows to achieve the desired state of the network: chimera state or coherent pattern, full or relay inter-layer synchronization.

Combining the delayed interactions with the multiplex framework considered in this work can provide additional insight into the formation of the complex spatio-temporal patterns in real-world systems. Specifically, in brain networks where EEG patterns are recently reported to display chimera-like behavior at the onset of a seizure [

JS did the numerical simulations and the theoretical analysis. AZ supervised the study. All authors designed the study and contributed to the preparation of the manuscript. All the authors have read and approved the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer—163436311—SFB 910 and by the German Academic Exchange Service (DAAD) and the Department of Science and Technology of India (DST) within the PPP project (INT/FRG/DAAD/P-06/2018).