Master Stability Function Research Articles

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.

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.

The linearity of the master stability function

The master stability function (MSF) is a tool to evaluate the local stability of the synchronization in coupled oscillators. Computing the MSF of a network of a specific oscillator results in a curve whose shape is dependent on the nodes' dynamics, network topology, coupling function, and coupling strength. This paper calculates the MSF of networks of two diffusively coupled oscillators by considering different single variable and multi-variable couplings. Then, the linearity of the MSF is investigated by fitting a straight line to the MSF curve, and the root mean square error is obtained. It is observed that the multi-variable coupling with equal coefficients on all variables results in a linear MSF regardless of the dynamics of the nodes.

Equivalent synchronization patterns in chaotic jerk systems

Jerk systems are some of the simplest dynamical systems that can exhibit chaotic dynamics. This paper investigates the synchronization of coupled jerk systems with coupling in single variables. We apply the well-known approach for synchronization analysis, the master stability function, which determines the stability of the synchronization manifold. It is shown that a jerk system in which the jerk equation is not dependent on the acceleration has similar master stability functions when coupled in velocity or acceleration variables. Therefore, the system has the same synchronization behavior in these two coupling configurations. Such an equivalence has not been reported in the literature.

Investigating Bifurcation Points of Complex Network Synchronization

This paper investigates the relationship between synchronization, system dynamics, and bifurcation points. To investigate the synchronization of dynamical systems, first, two oscillators are considered. Then networks with different structures are generated. The Master Stability Function (MSF) and synchronization criterion are computed for various oscillator dynamics in each network. The results of this paper can be categorized into two groups. In the first group, three points around the bifurcation points of the systems (before, near, and after the bifurcation point) are investigated that show no specific variation in the MSF diagram or its zero-crossing. Then it is revealed that the zero-crossing of the MSF diagram is related to the bifurcation points of the synchronization measure. In the second group, the types of synchronization transitions and MSFs for two periodic and chaotic dynamics of the studied systems are investigated for different networks. The results show that the type of synchronization transition can be different by changing the dynamics or network structure. However, in all networks and dynamics, the zero-crossing of MSF is completely matched with the bifurcation points of synchronization measure.

Master stability functions for metacommunities with two types of habitats.

Current questions in ecology revolve around instabilities in the dynamics on spatial networks and particularly the effect of node heterogeneity. We extend the master stability function formalism to inhomogeneous biregular networks having two types of spatial nodes. Notably, this class of systems also allows the investigation of certain types of dynamics on higher-order networks. Combined with the generalized modeling approach to study the linear stability of steady states, this is a powerful tool to numerically asses the stability of large ensembles of systems. We analyze the stability of ecological metacommunities with two distinct types of habitats analytically and numerically in order to identify several sets of conditions under which the dynamics can become stabilized by dispersal. Our analytical approach allows general insights into stabilizing and destabilizing effects in metapopulations. Specifically, we identify self-regulation and negative feedback loops between source and sink populations as stabilizing mechanisms and we show that maladaptive dispersal may be stable under certain conditions.

Effects of Amplitude, Maximal Lyapunov Exponent, and Kaplan–Yorke Dimension of Dynamical Oscillators on Master Stability Function

Obtaining the master stability function is a well-known approach to study the synchronization in networks of chaotic oscillators. This method considers a normalized coupling parameter which allows for a separation of network topology and local dynamics of the nodes. The present study aims to understand how the dynamics of oscillators affect the master stability function. In order to examine the effect of various characteristics of oscillators, a flexible oscillator with adjustable amplitude, Lyapunov exponent, and Kaplan–Yorke dimension is used. Not surprisingly, it is demonstrated that the amplitude of the oscillations has no substantial effect on the master stability function, i.e. the coupling strength needed for the complete synchronization is not changed. However, the flexible oscillators with larger maximal Lyapunov exponent synchronize with larger coupling strength. Interestingly, it is shown that there is no linear connection between the Kaplan–Yorke dimension and coupling strength needed for complete synchronization.

Matryoshka and disjoint cluster synchronization of networks.

The main motivation for this paper is to characterize network synchronizability for the case of cluster synchronization (CS), in an analogous fashion to Barahona and Pecora [Phys. Rev. Lett. 89, 054101 (2002)] for the case of complete synchronization. We find this problem to be substantially more complex than the original one. We distinguish between the two cases of networks with intertwined clusters and no intertwined clusters and between the two cases that the master stability function is negative either in a bounded range or in an unbounded range of its argument. Our proposed definition of cluster synchronizability is based on the synchronizability of each individual cluster within a network. We then attempt to generalize this definition to the entire network. For CS, the synchronous solution for each cluster may be stable, independent of the stability of the other clusters, which results in possibly different ranges in which each cluster synchronizes (isolated CS). For each pair of clusters, we distinguish between three different cases: Matryoshka cluster synchronization (when the range of the stability of the synchronous solution for one cluster is included in that of the other cluster), partially disjoint cluster synchronization (when the ranges of stability of the synchronous solutions partially overlap), and complete disjoint cluster synchronization (when the ranges of stability of the synchronous solutions do not overlap).

Coupling-induced periodic windows in networked discrete-time systems

Networked nonlinear systems present a variety of emergent phenomena as a result of the mutual interactions between their units. An interesting feature of these systems is the presence of stable periodic behavior even when each unit oscillates chaotically if in isolation. Surprisingly, the mechanism in which the network interaction replaces chaos by periodicity is still poorly understood. Here, we show that such an onset of regularity can occur via replication of periodic windows. This phenomenon multiplies the stability domains in the system parameter space, not only suppressing chaos but also making the network less vulnerable to external disturbances such as shocks and noise. Moreover, we observe that the network cluster synchronizes for the parameters corresponding to the replica periodic windows. To confirm these observations, we employ the formalism of the master stability function demonstrating that the complete synchronized state is indeed transversally unstable in the replica windows.

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.

Synchronization and pinning control of stochastic coevolving networks

Network dynamical systems are often characterized by the interlaced evolution of the node and edge dynamics, which are driven by both deterministic and stochastic factors. This manuscript offers a general mathematical model of coevolving network, which associates a state variable to each node and edge in the network, and describes their evolution through coupled stochastic differential equations. We study the emergence of synchronization, be it spontaneous or induced by a pinning control action, and provide sufficient conditions for local and global convergence. We enable the use of the Master Stability Function approach for studying coevolving networks, thereby obtaining conditions for almost sure local exponential convergence, whereas global conditions are derived using a Lyapunov-based approach. The theoretical results are then leveraged to design synchronization and pinning control protocols in two select applications. In the first one, the edge dynamics are tailored to induce spontaneous synchronization, whereas in the second the pinning edges are activated/deactivated and their weights modulated to drive the network towards the pinner’s trajectory in a distributed fashion.

Synchronizability of Multi-Layer Dual-Center Coupled Star Networks

In the research on complex networks, synchronizability is a significant measurement of network nature. Several research studies center around the synchronizability of single-layer complex networks and few studies on the synchronizability of multi-layer networks. Firstly, this paper calculates the Laplacian spectrum of multi-layer dual-center coupled star networks and multi-layer dual-center coupled star–ring networks according to the master stability function (MSF) and obtains important indicators reflecting the synchronizability of the above two network structures. Secondly, it discusses the relationships among synchronizability and various parameters, and numerical simulations are given to illustrate the effectiveness of the theoretical results. Finally, it is found that the two sorts of networks studied in this paper are of the same synchronizability, and compared with that of a single-center network structure, the synchronizability of two dual-center structures is relatively weaker.

Smallest Chimeras Under Repulsive Interactions.

We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions can only establish a symmetry of complete synchrony in the ring, which is broken with increasing repulsive interactions when the junctions pass through serials of asynchronous states (periodic and chaotic) but finally emerge into chimera states. The chimera pattern first appears in chaotic rotational motion of the three junctions when two junctions evolve coherently, while the third junction is incoherent. For larger repulsive coupling, the junctions evolve into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one junction transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions in the sense that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions, while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization, and two chimera patterns are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.

Relay interlayer synchronisation: invariance and stability conditions

In this paper, the existence (invariance) and stability (locally and globally) of relay interlayer synchronisation (RIS) are investigated in a chain of multiplex networks. The local dynamics of the nodes in the symmetric positions layers on both sides of the non-identical middlemost layer(s) are identical. The local and global stability conditions for this synchronisation state are analytically derived based on the master stability function approach and by constructing a suitable Lyapunov function, respectively. We propose an appropriate demultiplexing process for the existence of the RIS state. Then the variational equation transverse to the RIS manifold for demultiplexed networks is derived. In numerical simulations, the impact of interlayer and intralayer coupling strengths, variations of the system parameter in the relay layers and demultiplexing on the emergence of RIS in triplex and pentaplex networks are explored. Interestingly, in this multiplex network, enhancement of RIS is observed when a type of impurity via parameter mismatch in the local dynamics of the nodes is introduced in the middlemost layer. A common time-lag with small amplitude shift between the symmetric positions and central layers plays an important role for the enhancing of relay interlayer synchrony. This analysis improves our understanding of synchronisation states in multiplex networks with nonidentical layers.

Interaction between economies in a business cycle model

In this work we study the interaction between two or three nonlinear dynamical macroeconomic idealized systems. The model is a modified version of the Bouali’s system. Only three economic variables were considered: foreign capital inflow, household savings, and the gross domestic product. It was studied possible patterns that can result from these dynamics as well as structural changes when the coupling parameter (inflow of external capital) varies, here chosen as control parameter. Firstly, the present model has multistability where limit cycles and strange attractors depend on the initial conditions. Moreover, to some values of the control parameter occurs synchronization between the economies. However, there is a range of control parameters where they can be unstable. So, we applied the Master Stability Function method in order to evaluate the stability analysis. We found the occurrence of stable synchronization when the coupling of the economies is symmetrical (bidirectional interaction) or asymmetrical (unidirectional). Instability is predominant for higher control parameter values.

Group synchrony, parameter mismatches, and intragroup connections.

Group synchronization arises when two or more synchronization patterns coexist in a network formed of oscillators of different types, with the systems in each group synchronizing on the same time evolution, but systems in different groups synchronizing on distinct time evolutions. Group synchronization has been observed and characterized when the systems in each group are identical and the couplings between the systems satisfy specific conditions. By relaxing these constraints and allowing them to be satisfied in an approximate rather than exact way, we observe that stable group synchronization may still occur in the presence of small deviations of the parameters of the individual systems and of the couplings from their nominal values. We analyze this case and provide necessary and sufficient conditions for stability through a master stability function approach, which also allows us to quantify the synchronization error. We also investigate the stability of group synchronization in the presence of intragroup connections and for this case extend some of the existing results in the literature. Our analysis points out a broader class of matrices describing intragroup connections for which the stability problem can be reduced in a low-dimensional form.

Analysis of the synchronizability of two-layer chain networks with two inter-layer edges

The structure and dynamics of multi-layer networks are the frontiers in the current network science research. The inter-layer connections in a multi-layer network directly affect the dynamics, propagation, diffusion, and synchronization of the entire multi-layer network. However, there is rarely systematic theory or method for the general structure of multi-layer networks. People can only start with a simple structure to explore how the inter-layer connections affect the overall dynamic behaviors of multiple layers. The work of this paper aims at the two-layer networks where each layer is a chain network. Through graph theory and numerical calculation, this problem is explored to find the optimization method of the inter-layer connections. First of all, according to the master stability function method, this paper proves that when the synchronous region is unbounded, a multi-layer network with the same network structure at each layer has a certain inter-layer coupling strength threshold, so that the entire network reaches the maximum synchronizability, which is the synchronizability of its single-layer network. Secondly, the simulations show that when there are only two edges between layers, the optimal positions are respectively located at $1/4$ and $3/4$ of each chain, while the worst positions are the end node of the chain and its neighbor node. Furthermore, we discuss in detail about the optimal connection modes of two-layer chain networks when the number of single-layer nodes is $4N+x~(x=0,~1,~2,~3)$ using the master stability function method and “network shortest distance” structure index. In the case that the single-layer nodes number is $4N$, there are $4$ optimal connection modes; in the other three cases, there is only one optimal connection mode respectively. Finally, we propose the expression of the optimal inter-layer coupling strength, and then verify the correctness of the expression by calculating the optimal inter-layer coupling strength values of the two-layer chain network with different sizes.

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.

Collective Coherence Resonance in Networks of Optical Neurons

The dynamical properties of an optical neuron formed by a quantum dot semiconductor laser model subjected to optical injection and optical feedback are analyzed. The parameter space spanned by the injection strength and frequency detuning of the optical injection is systematically scanned and modulations of the bifurcation boundaries that induce complex scenarios are found, which enable new opportunities to introduce the optical setup as an optical neuron. The counterintuitive behavior of coherence resonance for different setups of a single‐driven optical neuron under optical feedback is also found. Following the results, the microscopically motivated quantum dot laser rate equation model is reduced to the normal form of a saddle‐node infinite period (SNIPER) bifurcation for low injection strengths and a network of four such SNIPER systems in a globally coupled setup is studied. A new phenomenon is observed, which is called collective coherence resonance. This novel dynamical behavior is connected to the coexistence of a network‐wide SNIPER bifurcation and a change in stability for the synchronized manifold, analyzed via the master stability function.

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.