Intralayer Synchronization Research Articles

Synchronization on fractional multiplex higher-order networks.

This paper explores the synchronization problem in fractional multiplex higher-order networks. Initially, a fractional multiplex higher-order network model is established, which seamlessly integrates multiplex structures with higher-order interactions. Subsequently, by leveraging a well-crafted Lyapunov function, the Lyapunov direct method, and fractional inequalities, it is demonstrated that the fractional multiplex higher-order network can achieve intra-layer synchronization, inter-layer synchronization, and complete synchronization. Finally, the theoretical findings are validated through two numerical examples featuring a simplicial complex or hypergraph structures within the intra-layer network.

Pinning intra-layer synchronization in multiplex networks of nonidentical layers

In this paper, we investigate intra-layer synchronization in multiplex networks via pinning control strategies. All layers have different topologies and different intrinsic node dynamics. Novel sufficient criteria are derived for pinning intra-layer synchronization of multiplex networks, which indicates that the pinned node set can stabilize the multiplex network if, for each layer, the pinned node(s) can access all unpinned nodes. A distributed adaptive algorithm and finite-time pinning algorithm for intra-layer synchronization of multiplex networks are also proposed. Finally, we give some numerical examples to validate the effectiveness of these theoretical results.

Pinning Control of Multiplex Dynamical Networks Using Spectral Graph Theory.

Pinning control has been attracting wide attention for the study of various complex networks for decades. This article explores grounded theory on the pinning synchronization of the emerging multiplex dynamical networks. The multiplex dynamical networks under study can describe many real-world scenarios, in which different layers have distinct individual dynamics of node. In this work, we build the bridge between multiplex structures and network dynamics by using the Lyapunov stability theory and the spectral graph theory. Furthermore, by analyzing spectral properties of the grounded super-Laplacian matrices, we set up several graph-based synchronization criteria for multiplex networks via pinning control. In addition, we overcome the difficulties induced by distinct node dynamics in different layers, and find that interlayer coupling strengths promote intralayer synchronization of multiplex networks. Finally, a collection of numerical simulations verifies the effectiveness of theoretical results.

Mixed synchronization in multiplex networks of counter-rotating oscillators

In this study, we introduce a novel concept known as ”mixed synchronization” within the context of two indirectly coupled counter-rotating oscillators. Our investigation begins by exploring the dynamics of two co-rotating oscillators when they are connected through a counter-rotating relay oscillator. Remarkably, this coupling mechanism leads to the emergence of what we term ”relay synchronization,” which signifies complete synchronization between the outer oscillators. This intriguing phenomenon lays the foundation for our subsequent exploration. We delve deeper into the dynamics of counter-rotating oscillators and discover that when two such oscillators are linked through relay oscillators, a distinct form of synchronization arises, which we aptly term ”relay mixed synchronization.” Notably, this reveals that even if counter-rotating oscillators are part of a larger network and indirectly coupled through a simple diffusive mechanism, it is possible to induce mixed synchronization. We extend the same innovative concept to the realm of multiplex networks, which consist of a finite number of counter-rotating oscillators in each layers and we explore both intralayer and interlayer mixed synchronization states. We verify the mixed synchronization state by master stability function analysis. We numerically verify the results by considering limit cycle and chaotic oscillators. This extension of our concept to multiplex networks showcases the versatility and applicability of mixed synchronization in diverse scenarios.

Intralayer Synchronization in Heterogeneous Multiplex Dynamical Networks Based on Spectral Graph Theory

Intralayer Synchronization in Heterogeneous Multiplex Dynamical Networks Based on Spectral Graph Theory

Synchronization Induced by Layer Mismatch in Multiplex Networks.

Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers.

Interlayer antisynchronization in degree-biased duplex networks.

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Event-Triggered Control for Intra/Inter-Layer Synchronization and Quasi-Synchronization in Two-Layer Coupled Networks

This paper studies the intra/inter-layer synchronization and quasi-synchronization in two-layer coupled networks via event-triggered control, in which different layers have mutually independent topologies. First, based on Lyapunov stability theory and event-triggered thoughts, hybrid controllers are designed, respectively, for intra-layer synchronization (ALS) and inter-layer synchronization (RLS). Second, a novel event-triggered rule is proposed, under which intra-layer quasi-synchronization (ALQS) and inter-layer quasi-synchronization (RLQS) can be respectively realized, and the event-triggered frequency can be greatly reduced. Moreover, the upper bound of the synchronization error can be flexibly adjusted by changing the parameters in event-triggered conditions, and the Zeno phenomenon about event-triggered control is also discussed in this paper. Finally, numerical examples are provided to confirm the correctness and validity of the proposed scheme.

Fixed-time synchronization of switched duplex networks with stochastic disturbances and limited communication

This paper investigates the issue of fixed-time intralayer synchronization of stochastic switched duplex networks (SSDNs) with limited communication. Firstly, control scheme without sign function is created to explore intralayer synchronization of SSDNs within a fixed time, and the control gains can vary with the error information. By the Lyapunov functional method, some novel synchronization criteria are established. In addition, by means of introducing the channel matrices, this paper considers the limited communication. Moreover, numerical simulation indicates the correctness of our theoretical result. Finally, from the perspective of the application, image encryption is demonstrated to confirm our theoretical implementation.

Dynamic event-triggered control for intra/inter-layer synchronization in multi-layer networks

In this paper, intra-layer synchronization (ALS) and inter-layer synchronization (RLS) of multi-layer networks (MLNs) are studied, where the topology of each layer is different and the inter-layer connection is node-to-node if two layers are connected. Dynamic event-triggered (DET) control is first introduced into the design of synchronization controllers in MLNs to reduce communication frequency. Based on the Lyapunov functional method, sufficient conditions for ALS and RLS are strictly derived by designing appropriate DET controllers. It is theoretically verified that the designed DET conditions in MLNs further reduce triggering frequency compared with static event-triggered (SET) conditions, and Zeno phenomenon can be eliminated. Finally, the validity of the theoretical results is illustrated by numerical simulations.

INTERLAYER AND INTRALAYER SYNCHRONIZATION IN MULTIPLEX FRACTIONAL-ORDER NEURONAL NETWORKS

Fractional-order models describing neuronal dynamics often exhibit better compatibility with diverse neuronal firing patterns that can be observed experimentally. Due to the overarching significance of synchronization in neuronal dynamics, we here study synchronization in multiplex neuronal networks that are composed of fractional-order Hindmarsh–Rose neurons. We compute the average synchronization error numerically for different derivative orders in dependence on the strength of the links within and between network layers. We find that, in general, fractional-order models synchronize better than integer-order models. In particular, we show that the required interlayer and intralayer coupling strengths for interlayer or intralayer synchronization can be weaker if we reduce the derivative order of the model describing the neuronal dynamics. Furthermore, the dependence of the interlayer or intralayer synchronization on the intralayer or interlayer coupling strength vanishes with decreasing derivative order. To support these results analytically, we use the master stability function approach for the considered multiplex fractional-order neuronal networks, by means of which we obtain sufficient conditions for the interlayer and intralayer synchronizations that are in agreement with numerical results.

Explosive synchronization in coupled nonlinear oscillators on multiplex network

We report the emergence of explosive synchronization in a multiplex network where oscillators on the first layer are coupled with attractive coupling and those on the second layer, coupled with repulsive coupling. With Stuart-Landau and FitzHugh-Nagumo oscillators as the nodal dynamics, we consider non-local and mean-field intralayer couplings. We establish that explosive synchronization occurs in the multiplex network in the presence of Gaussian noise, and the transition is first order with hysteresis to a state of complete intra-layer and in-phase interlayer synchronization. The width of the hysteresis depends on the range of intralayer attractive coupling, strength of interlayer coupling and noise. We also see how to have control over this induced transition so that the explosive nature, if undesirable, can be converted to continuous type by tuning the strength of repulsive coupling or noise.

Synchronization in a Multiplex Network of Nonidentical Fractional-Order Neurons

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron.

Intralayer and interlayer synchronization in multiplex network with higher-order interactions.

Recent developments in complex systems have witnessed that many real-world scenarios, successfully represented as networks, are not always restricted to binary interactions but often include higher-order interactions among the nodes. These beyond pairwise interactions are preferably modeled by hypergraphs, where hyperedges represent higher-order interactions between a set of nodes. In this work, we consider a multiplex network where the intralayer connections are represented by hypergraphs, called the multiplex hypergraph. The hypergraph is constructed by mapping the maximal cliques of a scale-free network to hyperedges of suitable sizes. We investigate the intralayer and interlayer synchronizations of such multiplex structures. Our study unveils that the intralayer synchronization appreciably enhances when a higher-order structure is taken into consideration in spite of only pairwise connections. We derive the necessary condition for stable synchronization states by the master stability function approach, which perfectly agrees with the numerical results. We also explore the robustness of interlayer synchronization and find that for the multiplex structures with many-body interaction, the interlayer synchronization is more persistent than the multiplex networks with solely pairwise interaction.

Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functions.

The stability analysis of synchronization patterns on generalized network structures is of immense importance nowadays. In this article, we scrutinize the stability of intralayer synchronous state in temporal multilayer hypernetworks, where each dynamic units in a layer communicate with others through various independent time-varying connection mechanisms. Here, dynamical units within and between layers may be interconnected through arbitrary generic coupling functions. We show that intralayer synchronous state exists as an invariant solution. Using fast-switching stability criteria, we derive the condition for stable coherent state in terms of associated time-averaged network structure, and in some instances we are able to separate the transverse subspace optimally. Using simultaneous block diagonalization of coupling matrices, we derive the synchronization stability condition without considering time-averaged network structure. Finally, we verify our analytically derived results through a series of numerical simulations on synthetic and real-world neuronal networked systems.

Intralayer synchronization in a duplex network with noise

This paper concerns the impact of environmental noise on the intralayer synchronization of the duplex networks. A duplex network contains two layers. Different from the previous works (Tang et al., 2018; Shen & Tang, 2018), environmental noise is introduced into the dynamical system of the duplex network. We incorporate both the interlayer delay and the intralayer delay into the dynamical system. Both of the delays are time‐varying. However, Tang et al. (2018) only considered the intralayer delays, and they were assumed as constants, while Shen and Tang (2018) did not consider the interlayer delay or intralayer delay. When the system does not achieve automatic intralayer synchronization, we introduce two controllers: one is the state‐feedback controller and the other is the adaptive state‐feedback controller. Interestingly, we find that the intralayer synchronization will be achieved automatically if the interlayer coupling strength is large enough and the intralayer coupling strength is small enough under the situation that the time‐varying interlayer delays are absent. Finally, some interesting simulation results are obtained for the Chua–Chua chaotic system withapplication of our theoretic results, which show the feasibility and effectiveness of our control schemes.

Relay Synchronization in a Weighted Triplex Network

Relay synchronization in multi-layer networks implies inter-layer synchronization between two indirectly connected layers through a relay layer. In this work, we study the relay synchronization in a three-layer multiplex network by introducing degree-based weighting mechanisms. The mechanism of within-layer connectivity may be hubs-repelling or hubs-attracting whenever low-degree or high-degree nodes receive strong influence. We adjust the remote layers to hubs-attracting coupling, whereas the relay layer may be unweighted, hubs-repelling, or hubs-attracting network. We establish that relay synchronization is improved when the relay layer is hubs-repelling compared to the other cases. We determine analytically necessary stability conditions of relay synchronization state using the master stability function approach. Finally, we explore the relation between synchronization and the topological property of the relay layer. We find that a higher clustering coefficient hinders synchronizability, and vice versa. We also look into the intra-layer synchronization in the proposed weighted triplex network and establish that intra-layer synchronization occurs in a wider range when relay layer is hubs-attracting.

Antiphase synchronization in multiplex networks with attractive and repulsive interactions.

A series of recent publications, within the framework of network science, have focused on the coexistence of mixed attractive and repulsive (excitatory and inhibitory) interactions among the units within the same system, motivated by the analogies with spin glasses as well as to neural networks, or ecological systems. However, most of these investigations have been restricted to single layer networks, requiring further analysis of the complex dynamics and particular equilibrium states that emerge in multilayer configurations. This article investigates the synchronization properties of dynamical systems connected through multiplex architectures in the presence of attractive intralayer and repulsive interlayer connections. This setting enables the emergence of antisynchronization, i.e., intralayer synchronization coexisting with antiphase dynamics between coupled systems of different layers. We demonstrate the existence of a transition from interlayer antisynchronization to antiphase synchrony in any connected bipartite multiplex architecture when the repulsive coupling is introduced through any spanning tree of a single layer. We identify, analytically, the required graph topologies for interlayer antisynchronization and its interplay with intralayer and antiphase synchronization. Next, we analytically derive the invariance of intralayer synchronization manifold and calculate the attractor size of each oscillator exhibiting interlayer antisynchronization together with intralayer synchronization. The necessary conditions for the existence of interlayer antisynchronization along with intralayer synchronization are given and numerically validated by considering Stuart-Landau oscillators. Finally, we also analytically derive the local stability condition of the interlayer antisynchronization state using the master stability function approach.

Interlayer Hebbian plasticity induces first-order transition in multiplex networks

Adaptation plays a pivotal role in the evolution of natural and artificial complex systems, and in the determination of their functionality. Here, we investigate the impact of adaptive interlayer processes on intra-layer synchronization in multiplex networks. The considered adaptation mechanism is governed by a Hebbian learning rule, i.e., the link weight between a pair of interconnected nodes is enhanced if the two nodes are in phase. Such adaptive coupling induces an irreversible first-order transition route to synchronization accompanied with a hysteresis. We provide rigorous analytic predictions of the critical coupling strengths for the onset of synchronization and de-synchronization, and verify all our theoretical predictions by means of extensive numerical simulations.

Enhancing synchrony in asymmetrically weighted multiplex networks

Abstract We uncover the transition scenarios for two different kinds of synchronization appearing in a multiplex network, namely intralayer and interlayer synchronization, considering different weighting mechanisms among the coupled oscillators. Particularly, we choose an unweighted Erdos-Renyi interaction topology and then the weights are associated to the edges depending on the ratio of the adjacent nodal degrees. While considering intralayer connectivity, whether stronger nodal influences are received from a high degree node or a low degree node, the networks are classified accordingly as hubs-attracting or hubs-repelling, respectively. The obtained synchronization phenomena considering these two networks are compared with that of the unweighted network. Our explorations reveal that the intralayer synchronization is greatly enhanced in case of hubs-attracting network whereas it is de-enhanced for hubs-repelling network compared to the unweighted network. The obtained numerical results are also verified by analytically deriving necessary stability conditions using master stability function approach. We also explain the synchrony phenomena through spectral analysis of the corresponding Laplacian matrices and find that higher heterogeneity among the input intensities of the nodes gives lesser synchronizability and vice-versa. Moreover, the sustainability of interlayer synchronization is also scrutinized against progressive removal of interlayer links following either increasing or decreasing degree sequence of the nodes.