Inter-layer Synchronization Research Articles

Synchronization in a Memristive Duplex Network: The Impact of Intra-layer and Inter-layer Synaptic Pathways

Synchronization in multiplex networks is crucial for modeling functional connectivity in computational neuroscience. This study analyzes synchronization in a memristive Hindmarsh-Rose duplex network, investigating how synaptic pathways affect synchronization. Initially, electrical, inner-linking, and chemical functions are separately applied to both intra- and inter-layer connections. Then, a more realistic approach combines intra-layer electrical and inner-linking functions with inter-layer chemical synapses. Field coupling is also examined as an inter-layer connection with different intra-layer synapses. Synchronization stability is evaluated using the master stability function and synchronization error. Results show that electrical and inner-linking functions support intra- and inter-layer synchronization as coupling strength increases. However, with chemical synapses, neurons achieve synchronization under limited conditions. Applying chemical synapses to inter-layer connections restricts inter-layer synchronization, although intra-layer synchronization remains observable with electrical or inner-linking functions. Field coupling effectively promotes inter-layer synchronization when paired with intra-layer chemical connections but reduces it with intra-layer electrical or inner-linking functions.

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.

On fixed-time interlayer synchronization of two-layer multiweighted complex dynamic networks: An economic and practical non-chattering adaptive control approach

This paper investigates the adaptive fixed-time (FxT) interlayer synchronization issue of two-layer complex dynamic networks (CDNs), where both multiple weights and interlayer links are considered. Above all, a new FxT stability lemma is given, which offers a more flexible condition and some more precise estimates of stabilized convergence time compared with the existing results. Subsequently, an economic and practical non-chattering continuous adaptive controller is designed, which is superior to the currently available ones by excluding the signum function and linear part and removing the parity restriction on the exponents in control law. Furthermore, by means of the Lyapunov method and the newly-established FxT stability theory, some novel FxT interlayer synchronization criteria are derived for the addressed two-layer multiweighted CDNs with the designed non-chattering continuous adaptive controller, and the synchronized settling time (SST) is also reckoned. It is found that the interlayer links weights can contribute to the interlayer synchronization, and the SST becomes smaller with the increase of interlayer link weight. Finally, the proposed results are validated by numerical examples.

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.

Inter-layer synchronization on a two-layer network of unified chaotic systems: The role of network nodal dynamics

The interplay between synchronization across layers and structures within layers still remains far unclear in multi-layer networks. Via our newly-constructed two-layer network consisting of the unified chaotic systems where the link within layer is linearly coupled and the link across layers is formed via sharing a common signal, this paper investigates the inter-layer synchronization on two-layer networks from the perspective of network nodal dynamics, as well as explore the influence of intra-layer network structures on inter-layer synchronization. Here we derive an analytical sufficient condition for inter-layer synchronization, which is only dependent on network nodal dynamics, showing a handful of Lorenz-type oscillators including the classic one are able to successfully realize inter-layer synchronization of the network for any connected intra-layer topological structures and diagonal coupling matrices. Further, four regular networks, small-world, scale-free and random networks are respectively used to run numerical simulations. Numerical results verify the presented theoretical results, and also show that Lü-type as well as Chen-type oscillators networks fail to realize inter-layer synchronization no matter which kinds of topologies and coupling matrices within layers. Specifically, for the case of identical topologies within layers, the chain-type and ring-type structures can stimulate larger range of Lorenz-type oscillators into synchronization across layers under the assumption of diagonal coupling matrices, but the fully-connected and star-type structures fail, and the non-diagonal intra-layer coupling matrix on the whole weaken the inter-layer synchronizability. For the case of nonidentical topologies within layers, the connection structure within layers can hardly enlarge the range of oscillators for synchronization across layers, even narrow the range, but some structures may enlarge the range of coupling strengthes for inter-layer 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.

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.

Finite Time Inter-Layer Synchronization of Duplex Networks via Event-Dependent Intermittent Control

This brief is devoted to the finite time inter-layer synchronization of duplex networks by means of intermittent control. Different from most previous works, the intermittent control scheme devised is an event-dependent type, which relaxes the strict constraints of control time. To be specific, the work time of intermittent controller is no longer set in advance, and determined by the relation between the designed Lyapunov function and two preset boundary functions. Based on the proposed intermittent control mechanism, a sufficient condition is given to guarantee the finite time inter-layer synchronization of the duplex networks. Finally, numerical examples are presented to illustrate the validness of theoretical results.

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.

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.

Edges of inter-layer synchronization in multilayer networks with time-switching links.

We investigate the transition to synchronization in a two-layer network of oscillators with time-switching inter-layer links. We focus on the role of the number of inter-layer links and the timescale of topological changes. Initially, we observe a smooth transition to complete synchronization for the static inter-layer topology by increasing the number of inter-layer links. Next, for a dynamic topology with the existent inter-layer links randomly changing among identical oscillators in the layers, we observe a significant improvement in the system synchronizability; i.e., the layers synchronize with lower inter-layer connectivity. More interestingly, we find that, for a critical switching time, the transition from the network state of low inter-layer synchronization to high inter-layer synchronization occurs abruptly as the number of inter-layer links increases. We interpret this phenomenon as shrinking and ultimately the disappearance of the basin of attraction of a desynchronized network state.

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.

Synchronization in multiplex networks of chaotic oscillators with frequency mismatch

We explore numerically inter-layer synchronization of spatiotemporal patterns in a heterogeneous bilayer network of nonlocally coupled Rössler oscillators. Both layers exhibit phase chimera regimes with similar cluster structures, but there is a mean frequency mismatch between the layers. We show that an entrainment of the mean frequency in the interacting layers is observed already for sufficiently weak inter-layer coupling even when the frequency mismatch is large. At the same time, frequency synchronization does not mean structure synchronization. Our numerical study indicates that inter-layer synchronization takes place within noticeably shorter ranges of the frequency mismatch and the inter-layer coupling strength than the frequency synchronization regime. Besides, we show that the weakest chaotic behavior is observed when both frequency and structure synchronizations occur, while the most developed chaotic oscillations are typical when the spatiotemporal structures in the layers are desynchronized.

Interlayer synchronization of duplex time-delay network with delayed pinning impulses

This paper mainly focuses on interlayer synchronization in a duplex network setting with the intralayer coupling delay and the interlayer transmission delay via pinning impulsive control. Firstly, based on Lyapunov stability theory, a general pinning impulsive strategy guaranteeing interlayer synchronization is obtained. Then, the minimum number of pinned nodes and the maximum impulse interval needed theoretically for interlayer synchronization are obtained. Moreover, when some parameters, including the impulse strengths, the ratio of pinned nodes and the impulse interval, are fixed certain values, their influences on the stable region are further discussed. It is found that the stable region becomes smaller as the ratio of pinned nodes decreases, and gets larger as the impulse strength with no delay is closer to −1 and that with delay is closer to 0. At the same time, the small intralayer coupling delay and interlayer transmission delay can enlarge the stable region. Finally, numerical simulations are provided to verify the effectiveness and correctness of our results.

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.

Finite-time inter-layer projective synchronization of Caputo fractional-order two-layer networks by sliding mode control

Finite-time inter-layer projective synchronization (FIPS) of Caputo fractional-order two-layer networks (FTN) based on sliding mode control (SMC) technique is investigated in this article. Firstly, in order to realize the FIPS of FTN, a fractional-order integral sliding mode surface (SMS) is established. Then, through the theory of SMC, we design a sliding mode controller (SMCr) to ensure the appearance of sliding mode motion. According to the fractional Lyapunov direct method, the trajectories of the system are driven to the proposed SMS, and some novel sufficient conditions for FIPS of FTN are derived. Furthermore, as two special cases of FIPS, finite-time inter-layer synchronization and finite-time inter-layer anti-synchronization for the FTN are studied. Finally, this paper takes the fractional-order chaotic Lü’s system and the fractional-order chaotic Chen’s system as the isolated node of the first layer system and the second layer system, respectively. And the numerical simulations are given to demonstrate the feasibility and validity of the proposed theoretical results.

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.

Synchronization Analysis on Two-Layer Networks of Fractional-Order Systems: IntraIayer and InterIayer Synchronization

The paper focuses on a two-layer network composed of fractional-order oscillators, and investigates the mesoscale collective dynamical behaviors of the network, i.e., intra-layer synchronization and inter-layer synchronization. Specifically, the necessary constraint conditions are firstly presented for achieving intra-layer synchronization and inter-layer synchronization. With the constraints, based on the Lyapunov stability theory and the fractional-order system theory, sufficient conditions for inter-layer synchronization and intra-layer synchronization are obtained respectively. Interestingly, the conditions cover the basic synchronization criteria bridging the relationship of interacting between the topologies, inner coupling functions and coupling strengthes of the intra-layer and the inter-layer. Finally, numerical examples are provided to further verify the theoretical results.

Interlayer adaptation-induced explosive synchronization in multiplex networks

It is known that intra-layer adaptive coupling among connected oscillators instigates explosive synchronization (ES) in multilayer networks. Taking an altogether different cue in the present work, we consider inter-layer adaptive coupling in a multiplex network of phase oscillators and show that the scheme gives rise to ES with an associated hysteresis irrespective of the network architecture of individual layers. The hysteresis is shaped by the inter-layer coupling strength and the frequency mismatch between the mirror nodes. We provide rigorous mean-field analytical treatment for the measure of global coherence and manifest they are in a good match with respective numerical assessments. Moreover, the analytical predictions provide a complete insight into how adaptive multiplexing suppresses the formation of a giant cluster, eventually giving birth to ES. The study will help in spotlighting the role of multiplexing in the emergence of ES in real-world systems represented by multilayer architecture. Particularly, it is relevant to those systems which have limitations towards change in intra-layer coupling strength.