Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators

Node-layer duality in networked systems

Real-world networks typically exhibit several aspects, or layers, of interactions among their nodes. By permuting the role of the nodes and the layers, we establish a new criterion to construct the dual of a network. This approach allows to examine connectivity from either a node-centric or layer-centric viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrate that nodewise and layerwise connectivity measure different but related aspects of the same system. Leveraging node-layer duality provides complementary insights, enabling a deeper comprehension of diverse networks across social science, technology and biology. Taken together, these findings reveal previously unappreciated features of complex systems and provide a fresh tool for delving into their structure and dynamics.

Fiedler value: The cumulated dynamical contribution value of all edges in a complex network

Fiedler value, as the minimal real part of (or the minimal) nonzero Laplacian eigenvalue, garners significant attention as a metric for evaluating network topology and its dynamics. In this paper, we address the quantification relation between Fiedler value and each edge in a directed complex network, considering undirected networks as a special case. We propose an approach to measure the dynamical contribution value of each edge. Interestingly, these contribution values can be both positive and negative, which are determined by the left and right Fiedler vectors. Further, we show that the cumulated dynamical contribution value of all edges is exactly the Fiedler value. This provides a promising angle on the Fiedler value in terms of dynamics and network structure. Therefore, the percentage of contribution of each edge to the Fiedler value is quantified. Numerical results reveal that network dynamics is significantly influenced by a small fraction of edges, say, one single directed edge contributes to over 90% of the Fiedler value in the Cat Cerebral Cortex network.

Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapses

This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

Synchronization of non-smooth chaotic systems via an improved reservoir computing

The reservoir computing (RC) is increasingly used to learn the synchronization behavior of chaotic systems as well as the dynamical behavior of complex systems, but it is scarcely applied in studying synchronization of non-smooth chaotic systems likely due to its complexity leading to the unimpressive effect. Here proposes a simulated annealing-based differential evolution (SADE) algorithm for the optimal parameter selection in the reservoir, and constructs an improved RC model for synchronization, which can work well not only for non-smooth chaotic systems but for smooth ones. Extensive simulations show that the trained RC model with optimal parameters has far longer prediction time than those with empirical and random parameters. More importantly, the well-trained RC system can be well synchronized to its original chaotic system as well as its replicate RC system via one shared signal, whereas the traditional RC system with empirical or random parameters fails for some chaotic systems, particularly for some non-smooth chaotic systems.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.

Stability and synchronization in neural network with delayed synaptic connections

In this paper, we investigate the stability of the synchronous state in a complex network using the master stability function technique. We use the extended Hindmarsh-Rose neuronal model including time delayed electrical, chemical, and hybrid couplings. We find the corresponding master stability equation that describes the whole dynamics for each coupling mode. From the maximum Lyapunov exponent, we deduce the stability state for each coupling mode. We observe that for electrical coupling, there exists a mixing between stable and unstable states. For a good setting of some system parameters, the position and the size of unstable areas can be modified. For chemical coupling, we observe difficulties in having a stable area in the complex plane. For hybrid coupling, we observe a stable behavior in the whole system compared to the case where these couplings are considered separately. The obtained results for each coupling mode help to analyze the stability state of some network topologies by using the corresponding eigenvalues. We observe that using electrical coupling can involve a full or partial stability of the system. In the case of chemical coupling, unstable states are observed whereas in the case of hybrid interactions a full stability of the network is obtained. Temporal analysis of the global synchronization is also done for each coupling mode, and the results show that when the network is stable, the synchronization is globally observed, while in the case when it is unstable, its nodes are not globally synchronized.

Piezoelectric energy harvesting and synchronization of excited and modified Huygens's pendulums

We consider a model of modified Huygens pendulums in order to be able to study the dynamics of such a system and carry out piezoelectric energy harvesting and the effects of phenomena encountered on this energy harvesting. The modifications made to the system here are the use of compound pendulums, a parametric force, and the addition of a piezoelectric transducer for energy harvesting. Thanks to the Lagrangian formalism, the governing equations were established and the numerical resolution was made using the fourth-order Runge-Kutta algorithm. We observed the presence of several types of synchronization (in-phase, anti-phase, quadrature-phase) and the existence of periodic, multi-periodic, or chaotic dynamics. Also, synchronization plays an important role in energy harvesting, in particular, in-phase synchronization, which promises much better performance than anti-phase synchronization. The effects of system parameters (amplitude and frequency of parametric force, stiffness coefficient, electromechanical coupling coefficient, etc.) are also studied on synchronization and energy harvesting. These results have applications in the manufacture of sensors and actuators, the power supply of electronic devices, and the manufacture of autonomous devices.

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.

Fixed-time synchronization of delayed memristive neural networks with impulsive effects via novel fixed-time stability theorem

In this study, the fixed-time synchronization (FXTS) of delayed memristive neural networks (MNNs) with hybrid impulsive effects is explored. To investigate the FXTS mechanism, we first propose a novel theorem about the fixed-time stability (FTS) of impulsive dynamical systems, where the coefficients are extended to functions and the derivatives of Lyapunov function (LF) are allowed to be indefinite. After that, we obtain some new sufficient conditions for achieving FXTS of the system within a settling-time using three different controllers. At last, to verify the correctness and effectiveness of our results, a numerical simulation was conducted. Significantly, the impulse strength studied in this paper can take different values at different points, so it can be regarded as a time-varying function, unlike those in previous studies (the impulse strength takes the same value at different points). Hence, the mechanisms in this article are of more practical applicability.

Identification of disease propagation paths in two-layer networks

To determine the path of disease in different types of networks, a new method based on compressive sensing is proposed for identifying the disease propagation paths in two-layer networks. If a limited amount of data from network nodes is collected, according to the principle of compressive sensing, it is feasible to accurately identify the path of disease propagation in a multilayer network. Experimental results show that the method can be applied to various networks, such as scale-free networks, small-world networks, and random networks. The impact of network density on identification accuracy is explored. The method could be used to aid in the prevention of disease spread.

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.

Occurrence of super-diffusion in two-layer networks

Super-diffusion is a phenomenon that can be observed in multilayer networks, which describes that the diffusion in a multilayer network is faster than that in the fastest individual layer. In most studies of super-diffusion on two-layer networks, many researchers have focused on the overlap of edges in the two layers and the mode of interlayer connectivity. We discover that the occurrence of super-diffusion in two-layer networks is not necessarily related to the overlap degree. In particular, in a two-layer network, sparse topological structures of individual layers are more beneficial to the occurrence of super-diffusion than dense topological structures. Additionally, similar diffusion abilities of both layers favor super-diffusion. The density of interlayer edges and interlayer connection patterns also influence the occurrence of super-diffusion. This paper offers suggestions to improve the diffusion ability in two-layer networks, which can facilitate the selection of practical information transmission paths between different systems and optimize the design of the internal framework of a company composed of multiple departments.

Synchronizing network systems in the presence of limited resources via edge snapping

In this work, we propose a multilayer control protocol for the synchronization of network dynamical systems under limited resources. In addition to the layer where the interactions of the system take place, i.e., the backbone network, we propose a second, adaptive layer, where the edges are added or removed according to the edge snapping mechanism. Different from classic edge snapping, the inputs to the edge dynamics are modified to cap the number of edges that can be activated. After studying the local stability of the overall network dynamics, we illustrate the effectiveness of the approach on a network of Rössler oscillators and then show its robustness in a more general setting, exemplified with a model of the Italian high-voltage power grid.

Cluster synchronization induced by manifold deformation

Pinning control of cluster synchronization in a globally connected network of chaotic oscillators is studied. It is found in simulations that when the pinning strength exceeds a critical value, the oscillators are synchronized into two different clusters, one formed by the pinned oscillators and the other one formed by the unpinned oscillators. The numerical results are analyzed by the generalized method of master stability function (MSF), in which it is shown that whereas the method is able to predict the synchronization behaviors of the pinned oscillators, it fails to predict the synchronization behaviors of the unpinned oscillators. By checking the trajectories of the oscillators in the phase space, it is found that the failure is attributed to the deformed synchronization manifold of the unpinned oscillators, which is clearly deviated from that of isolated oscillator under strong pinnings. A similar phenomenon is also observed in the pinning control of cluster synchronization in a complex network of symmetric structures and in the self-organized cluster synchronization of networked neural oscillators. The findings are important complements to the generalized MSF method and provide an alternative approach to the manipulation of synchronization behaviors in complex network systems.

Synchronization in a multilevel network using the Hamilton-Jacobi-Bellman (HJB) technique

This paper presents the optimal control and synchronization problem of a multilevel network of Rössler chaotic oscillators. Using the Hamilton-Jacobi-Bellman technique, the optimal control law with a three-state variable feedback is designed such that the trajectories of all the Rössler oscillators in the network are optimally synchronized at each level. Furthermore, we provide numerical simulations to demonstrate the effectiveness of the proposed approach for the cases of one and three networks. A perfect correlation between the MATLAB and PSpice results was obtained, thus allowing the experimental validation of our designed controller and shows the effectiveness of the theoretical results.

When multilayer links exchange their roles in synchronization

Real world networks contain multiple layers of links whose interactions can lead to extraordinary collective dynamics, including synchronization. The fundamental problem of assessing how network topology controls synchronization in multilayer networks remains open due to serious limitations of the existing stability methods. Towards removing this obstacle, we propose an approximation method which significantly enhances the predictive power of the master stability function for stable synchronization in multilayer networks. For a class of saddle-focus oscillators, including Rössler and piecewise linear systems, our method reduces the complex stability analysis to simply solving a set of linear algebraic equations. Using the method, we analytically predict surprising effects due to multilayer coupling. In particular, we prove that two coupling layers-one of which would alone hamper synchronization and the other would foster it-reverse their roles when used in a multilayer network. We also analytically demonstrate that increasing the size of a globally coupled layer, that in isolation would induce stable synchronization, makes the multilayer network unsynchronizable.

Partial locking in phase-oscillator populations with heterogenous coupling

We consider a variant of the mean-field model of coupled phase oscillators with uniform distribution of natural frequencies. By establishing correlations between the quenched disorder of intrinsic frequencies and coupling strength with both in- and out-coupling heterogeneities, we reveal a generic criterion for the onset of partial locking that takes place in a domain with the coexistence of phase-locked oscillators and drifters. The critical points manifesting the instability of the stationary states are obtained analytically. In particular, the bifurcation mechanism of the equilibrium states is uncovered by the use of frequency-dependent version of the Ott-Antonsen reduction consistently with the analysis based on the self-consistent approach. We demonstrate that both the manner of coupling heterogeneity and correlation exponent have influence on the emergent patterns of partial locking. Our research could find applicability in better understanding the phase transitions and related collective phenomena involving synchronization control in networked systems.

Fixed-Time Synchronization of Complex Dynamical Networks: A Novel and Economical Mechanism

Fixed-time synchronization of complex networks is investigated in this article. First, a completely novel lemma is introduced to prove the fixed-time stability of the equilibrium of a general ordinary differential system, which is less conservative and has a simpler form than those in the existing literature. Then, sufficient conditions are presented to realize synchronization of a complex network (with a target system) within a settling time via three different kinds of simple controllers. In general, controllers designed to achieve fixed-time stability consist of three terms and are discontinuous. However, in our mechanisms, the controllers only contain two terms or even one term and are continuous. Thus, our controllers are simpler and of more practical applicability. Finally, three examples are provided to illustrate the correctness and effectiveness of our results.

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.

A model for analyzing competitive dynamics on triplex networks

This paper studies the evolution process of competitive dynamics on triplex complex networks. We propose a new triplex network model in which the state of the node in each layer is affected by its neighbors as well as inter-layer competition. Through this model, we combine the opinion diffusion model, the Ising model, and the signed network and extend their application from single-layer to multi-layer networks. We derive the evolution process and dynamical equations of the model and carry out a series of numerical simulations to discuss the influence of several factors on the evolution process and the competitiveness of the network. First, we find that the increase of global transition threshold p or the proportion of initial active nodes will lead to more surviving layers and more active nodes in each layer. In addition, we summarize the similarities and differences of the evolution curves under different conditions. Second, we discuss the influence of initial active nodes and the average degree on the competitiveness of the network and find the correlations between them. Finally, we study the relationship between network topology and network competitiveness and conclude the conditions for the best competitiveness of the network. Based on the simulation results, we give specific suggestions on how to improve the competitiveness of the platform in reality.

Synchronizability of Multi-Layer-Coupled Star-Composed Networks

In this paper, several multi-layer-coupled star-composed networks with similar symmetrical structures are defined by using the theory of graph operation. The supra-Laplacian matrix of the corresponding multi-layer networks is obtained according to the master stability equation (MSF). Two important indexes that reflect the synchronizability of these kinds of networks are derived in the case of bounded and unbounded synchronized regions. The relationships among the synchronizability, the number of layers, the length of the paths, the branchings, and the interlayer and intralayer coupling strengths in the two cases are studied. At the same time, the simulation experiments are carried out with the MATLAB software, and the simulated images of the two symmetrical structure networks’ synchronizability are compared. Finally, the factors affecting the synchronizability of multi-layer-coupled star-composed networks are found. On this basis, optimization schemes are given to improve the synchronizability of multi-layer-coupled star-composed networks and the influences of the number of central nodes on the networks’ synchronizability are further studied.

Synchronizability of two-layer correlation networks

This study investigates the synchronizability of a typical type of two-layer correlation networks formed by two regular networks interconnected with two interlayer linking patterns, namely, positive correlation (PC) and negative correlation (NC). To analyze the network's stability, we consider the analytical expressions of the smallest non-zero and largest eigenvalues of the (weighted) Laplacian matrix as well as the linking strength and the network size for two linking patterns. According to the master stability function, the linking patterns, the linking strength, and the network size associated with two typical synchronized regions exhibit a profound influence on the synchronizability of the two-layer networks. The NC linking pattern displays better synchronizability than the PC linking pattern with the same set of parameters. Furthermore, for the two classical synchronized regions, the networks have optimal intralayer and interlayer linking strengths that maximize the synchronizability while minimizing the required cost. Finally, numerical results verify the validity of the theoretical analyses. The findings based on the representative two-layer correlation networks provide the basis for maximizing the synchronizability of general multiplex correlation networks.

Effect of magnetic induction on the synchronizability of coupled neuron network

Master stability functions (MSFs) are significant tools to identify the synchronizability of nonlinear dynamical systems. For a network of coupled oscillators to be synchronized, the corresponding MSF should be negative. The study of MSF will normally be discussed considering the coupling factor as a control variable. In our study, we considered various neuron models with electromagnetic flux induction and investigated the MSF's zero-crossing points for various values of the flux coupling coefficient. Our numerical analysis has shown that in all the neuron models we considered, flux coupling has increased the synchronization of the coupled neuron by increasing the number of zero-crossing points of MSFs or by achieving a zero-crossing point for a lesser value of a coupling parameter.

Superdiffusion criteria on duplex networks

Diffusion processes widely exist in nature. Some recent papers concerning diffusion processes focus their attention on multiplex networks. Superdiffusion, a phenomenon by which diffusion processes converge to equilibrium faster on multiplex networks than on single networks in isolation, may emerge because diffusion can occur both within and across layers. Some studies have shown that the emergence of superdiffusion depends on the topology of multiplex networks if the interlayer diffusion coefficient is large enough. This paper proposes some superdiffusion criteria relating to the Laplacian matrices of the two layers and provides a construction mechanism for generating a superdiffusible two-layered network. The method we proposed can be used to guide the discovery and construction of superdiffusible multiplex networks without calculating the second smallest Laplacian eigenvalues.

Synchronizability of double-layer dumbbell networks

Synchronization of multiplex networks has been a topical issue in network science. Dumbbell networks are very typical structures in complex networks which are distinguished from both regular star networks and general community structures, whereas the synchronous dynamics of a double-layer dumbbell network relies on the interlink patterns between layers. In this paper, two kinds of double-layer dumbbell networks are defined according to different interlayer coupling patterns: one with the single-link coupling pattern between layers and the other with the two-link coupling pattern between layers. Furthermore, the largest and smallest nonzero eigenvalues of the Laplacian matrix are calculated analytically and numerically for the single-link coupling pattern and also obtained numerically for the two-link coupling pattern so as to characterize the synchronizability of double-layer dumbbell networks. It is shown that interlayer coupling patterns have a significant impact on the synchronizability of multiplex systems. Finally, a numerical example is provided to verify the effectiveness of theoretical analysis. Our findings can facilitate company managers to select optimal interlayer coupling patterns and to assign proper parameters in terms of improving the efficiency and reducing losses of the whole team.

Random heterogeneity outperforms design in network synchronization

A widely held assumption on network dynamics is that similar components are more likely to exhibit similar behavior than dissimilar ones and that generic differences among them are necessarily detrimental to synchronization. Here, we show that this assumption does not generally hold in oscillator networks when communication delays are present. We demonstrate, in particular, that random parameter heterogeneity among oscillators can consistently rescue the system from losing synchrony. This finding is supported by electrochemical-oscillator experiments performed on a multielectrode array network. Remarkably, at intermediate levels of heterogeneity, random mismatches are more effective in promoting synchronization than parameter assignments specifically designed to facilitate identical synchronization. Our results suggest that, rather than being eliminated or ignored, intrinsic disorder in technological and biological systems can be harnessed to help maintain coherence required for function.

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.

Desynchronization Transitions in Adaptive Networks

Adaptive networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability approach for a large class of adaptive networks. This approach allows for reducing the synchronization problem for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. We show how the interplay between adaptivity and network structure gives rise to the formation of stability islands. Moreover, we report a desynchronization transition and the emergence of complex partial synchronization patterns induced by an increasing overall coupling strength. We illustrate our findings using adaptive networks of coupled phase oscillators and FitzHugh-Nagumo neurons with synaptic plasticity.

Dynamics of multilayer networks with amplification

We study the dynamics of a multilayer network of chaotic oscillators subject to amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster, and chimera states. Here, we consider a network with two layers of Rössler chaotic oscillators as well as applications to multilayer networks of the chaotic jerk and Liénard oscillators. Intra-layer coupling is considered to be all to all in the case of Rössler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Liénard, inter-layer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in-depth study on the case of a network of Rössler oscillators using a master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster, and phase synchronization with amplification. For the case of Rössler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification, and complete synchronization with amplification.

Symmetries and cluster synchronization in multilayer networks

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure to the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling.

Effect of topology upon relay synchronization in triplex neuronal networks

Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer networks. We show that including randomness in the connectivity structure either of the remote network layers or of the relay layer increases the range of interlayer coupling strength where relay synchronization can be observed.

Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks

We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuroscience and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.

Invariance and stability conditions of interlayer synchronization manifold

We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

Cluster synchronization: From single-layer to multi-layer networks

Cluster synchronization is a very common phenomenon occurring in single-layer complex networks, and it can also be observed in many multilayer networks in real life. In this paper, we study cluster synchronization of an isolated network and then focus on that of the network when it is influenced by an external network. We mainly explore how the influence layer impacts the cluster synchronization of the interest layer in a multilayer network. Considering that the clusters are changeable, we introduce a term called "cluster synchronizability" to measure the ability of a network to reach cluster synchronization. Since cluster synchronizability is intimately associated with the structure of the coupled external layer, we consider community networks and networks with different densities as the coupled layer. Besides the topology structure, the connection between two layers may also have an influence on the cluster synchronization of the interest layer. We study three different patterns of connection, including typical positive correlation, negative correlation, and random correlation and find that they all have a certain influence. However, the general theoretical analysis of cluster synchronization on multilayer networks is still a challenging topic. In this paper, we mainly use numerical simulations to discuss cluster synchronization.