Master Stability Function Research Articles

Similar Master Stability Functions for Different Coupling Schemes in Basic Chaotic Systems

Synchronization is a prominent phenomenon in coupled chaotic systems. The master stability function (MSF) is an approach that offers the prerequisites for the stability of complete synchronization, which is dependent on the coupling configuration. In this paper, some basic chaotic systems with the general form of the Sprott-A, Sprott-B, Sprott-D, Sprott-F, Sprott-G, Sprott-O, and Jerk systems are considered. For each system, their parametric form is designed, and constraints required to have similar MSFs in different coupling schemes are determined. Then, the parameters of the designed chaotic systems are found through an exhaustive computer search seeking chaotic solutions. The simplest cases found in this way are introduced, and similar synchronization patterns are confirmed numerically.

Helping networks to get synchronized: Effect of external stimulation

The attempts to evaluate the synchronizability of chaotic systems have shown that additive periodic forcing, as a relevant source of stimuli, significantly impacts multiple forms of synchrony. This paper investigates the complete synchronizability of coupled periodically forced chaotic systems using the master stability function method. Three classic chaotic systems, Lorenz, Chen's, and Hindmarsh-Rose models are employed for this study. The numerical simulations supporting master stability function findings are also reported. The impacts of forcing amplitude and frequency on the critical value of coupling strength at which synchronization occurs are determined. Evidence implies that, as the stimulation is amplified, the chaotic forced systems tend to synchronize at weaker couplings than the autonomous versions. In contrast, high-frequency stimulation is entirely ineffective. The required forcing amplitude is also relative to the system's attractor size.

Synchronization and multistability in a network of diffusively coupled laser models

Synchronization phenomena refer to the emergence of common temporal patterns among clusters of interacting units in a complex network. In information transmission, lasers’ synchronization plays a key role in facilitating information communication. This paper is devoted to studying the synchronization of globally coupled identical laser models via the linear and nonlinear forms of diffusive couplings. In this regard, the master stability function and time-averaged synchronization error are employed as the analytical and numerical approaches to examine complete synchronization. Apart from the complete synchrony, the constructed networks are explored to find other synchronization patterns and multistability. The results obtained from the master stability function analysis, which are further approved by the numerical outcomes, show that when coupled through the linear diffusive function, the interacting laser models achieve complete synchrony in a small value of the coupling parameter. However, in the nonlinear case, complete synchronization cannot be attained. Moreover, multistability can be observed in different network states, including cluster synchronization, chimera, and solitary states.

Effect of fractional derivatives on amplitude chimeras and symmetry-breaking death states in networks of limit-cycle oscillators.

We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.

Dynamical and topological conditions triggering the spontaneous activation of Izhikevich neuronal networks

Understanding the dynamic behavior of neuronal networks in silico is crucial for tackling the analysis of their biological counterparts and making accurate predictions. Of particular importance is determining the structural and dynamical conditions necessary for a neuronal network to activate spontaneously, transitioning from a quiescent ensemble of neurons to a network-wide coherent burst. Drawing from the versatility of the Master Stability Function, we have applied this formalism to a system of coupled neurons described by the Izhikevich model to derive the required conditions for activation. These conditions are expressed as a critical effective coupling gC, grounded in both topology and dynamics, above which the neuronal network will activate. For regular spiking neurons, average connectivity and noise play a significant role in their ability to activate. We have tested these conditions against numerical simulations of in silico networks, including both synthetic topologies and a biologically-realistic spatial network, showing that the theoretical conditions are well satisfied. Our findings indicate that neuronal networks readily meet the criteria for spontaneous activation, and that this capacity is weakly dependent on the microscopic details of the network as long as average connectivity and noise are sufficiently strong.

Master-Slave Outer Synchronization in Different Inner-Outer Coupling Network Topologies.

In this work, the problem of master-slave outer synchronization in different inner-outer network topologies is presented. Specifically, the studied inner-outer network topologies are coupled in master-slave configuration, where some particular scenarios concerning inner-outer topologies are addressed in order to disclose a suitable coupling strength to achieve outer synchronization. The novel MACM chaotic system is used as a node in the coupled networks, which presents robustness in its bifurcation parameters. Extensive numerical simulations are presented where the stability of the inner-outer network topologies is analyzed through a master stability function approach.

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.

Synchronization of stochastic complex networks with time-delayed coupling

Noise and time delay are inevitable in real-world networks. In this article, the framework of master stability function is generalized to stochastic complex networks with time-delayed coupling. The focus is on the effects of noise, time delay, and their inner interactions on the network synchronization. It is found that when there exists time-delayed coupling in the network and noise diffuses through all state variables of nodes, appropriately increasing the noise intensity can effectively improve the network synchronizability; otherwise, noise can be either beneficial or harmful. For stochastic networks, large time delays will lead to desynchronization. These findings provide valuable references for designing optimal complex networks in practical applications.

Master stability function for piecewise smooth Filippov networks

We consider a network of identical piecewise smooth bimodal systems, also known as systems of Filippov type, that synchronizes along the asymptotically stable periodic orbit of a single agent. We explicitly characterize the fundamental matrix solution of the network along the synchronous solution and extend the Master Stability Function tool to the present case of non-smooth dynamics of Filippov type.

Optimal time-varying coupling function can enhance synchronization in complex networks.

In this paper, we propose a time-varying coupling function that results in enhanced synchronization in complex networks of oscillators. The stability of synchronization can be analyzed by applying the master stability approach, which considers the largest Lyapunov exponent of the linearized variational equations as a function of the network eigenvalues as the master stability function. Here, it is assumed that the oscillators have diffusive single-variable coupling. All possible single-variable couplings are studied for each time interval, and the one with the smallest local Lyapunov exponent is selected. The obtained coupling function leads to a decrease in the critical coupling parameter, resulting in enhanced synchronization. Moreover, synchronization is achieved faster, and its robustness is increased. For illustration, the optimum coupling function is found for three networks of chaotic Rössler, Chen, and Chua systems, revealing enhanced synchronization.

Synchronization in repulsively coupled oscillators.

A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical oscillators. We illustrate the proposed coupling scheme on several paradigmatic two-coupled chaotic oscillators and validate its effectiveness through the linear stability analysis of the synchronous solution based on the master stability function approach. We further demonstrate that the proposed general condition for the selection of coupling profiles to achieve synchronization even works perfectly for a large ensemble of oscillators.

Synchronizability of multilayer star-ring networks with variable coupling strength

<abstract><p>We investigate the synchronizability of multilayer star-ring networks. Two types of multilayer networks, including aggregated coupling and divergent coupling, are established based on the connections between the hub node and the leaf nodes in the subnetwork. The eigenvalue spectrum of the two types of multilayer networks is strictly derived, and the correlation between topological parameters and synchronizability is analyzed by the master stability function framework. Moreover, the variable coupling strength has been investigated, revealing that it is significantly related to the synchronizability of the aggregated coupling while having no influence on the divergent coupling. Furthermore, the validity of the synchronizability analysis is obtained by implementing adaptive control on the multilayer star-ring networks previously mentioned. Calculations and comparisons show that the differences caused by the sizes of multilayer networks and interlayer coupling strength are not negligible. Finally, numerical examples are also provided to validate the effectiveness of the theoretical analysis.</p></abstract>

Dynamical effects of hypergraph links in a network of fractional-order complex systems.

In recent times, the fractional-order dynamical networks have gained lots of interest across various scientific communities because it admits some important properties like infinite memory, genetic characteristics, and more degrees of freedom than an integer-order system. Because of these potential applications, the study of the collective behaviors of fractional-order complex networks has been investigated in the literature. In this work, we investigate the influence of higher-order interactions in fractional-order complex systems. We consider both two-body and three-body diffusive interactions. To elucidate the role of higher-order interaction, we show how the network of oscillators is synchronized for different values of fractional-order. The stability of synchronization is studied with a master stability function analysis. Our results show that higher-order interactions among complex networks help the earlier synchronization of networks with a lesser value of first-order coupling strengths in fractional-order complex simplices. Besides that, the fractional-order also shows a notable impact on synchronization of complex simplices. For the lower value of fractional-order, the systems get synchronized earlier, with lesser coupling strengths in both two-body and three-body interactions. To show the generality in the outcome, two neuron models, namely, Hindmarsh-Rose and Morris-Leccar, and a nonlinear Rössler oscillator are considered for our analysis.

Symmetric synchronization behavior of multistable chaotic systems and circuits in attractive and repulsive couplings

This paper studies the synchronization behavior of multistable chaotic systems with coexisting symmetric attractors. Specifically, the focus is on the attractive and repulsive couplings of the attractors in single-variable couplings. It is shown that in the self-couplings, both attractors have the same synchronization pattern either in the attractive or repulsive coupling. In the cross-couplings, the synchronization pattern of the attractors is dependent on the variables involved in the coupling and the symmetry transformation. If the coupling scheme is defined such that only one of the variables in the coupling participates in the symmetry transformation, the synchronization patterns of the symmetric attractors are symmetric in the attractive and repulsive couplings. The master stability function is applied to four chaotic systems with different symmetry transformations to represent the results. The corresponding chaotic circuit of two coupled symmetric systems is also implemented and their symmetric responses are shown.

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.

When switching makes impossible synchronization possible

Synchronization of dynamical networks is of great significance in diverse scientific fields. In this paper, the focus is on the networks in which the synchronization stability region is bounded. We represent that blinking of determined links can lead to reaching a synchronous state for specific asynchronous networks. The requisite for the primary network is that it must have only one normalized coupling parameter (the first or the last) out of the stability region found by the master stability function. Then, the proper links that their substitution causes the other coupling parameter (the last or the first) to fall out of the stability region should be recognized. Finally, switching the identified links periodically can synchronize the network. This phenomenon is represented on two networks of Rӧssler and Lorenz oscillators.

Synchronization and bifurcation in an economic model.

We study the synchronization of two coupled idealized economies. In the present work, we consider a recently developed economic system that shows a richness of dynamical behavior. By means of the Lyapunov exponents, we analyze that there is overly complex behavior in the transitions in the dynamics of an isolated economy, oscillating between chaotic attractors and limit cycles. Then, for two coupled economies, we analyze the synchronization states for the space of all control parameters as a function of the network coupling parameter. Interestingly, we have evidenced that there is a broad region of fully synchronized states and as we increase the coupling, some phenomena such as a smooth and intermittent loss in synchronization emerge. In the same way, we observe phase synchronization for one of the control parameters. Ultimately, in order to confirm this loss of synchronization, we inspect the stability of synchronized states through the master stability function method for some control parameters. Here, we corroborate what was previously observed, the unexpected vast range of control parameter values of instability corresponding to desynchronization.

Synchronizability of Multilayer Directed Dutch Windmill Networks

This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.

Stability of synchronization in simplicial complexes with multiple interaction layers.

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

The structure of uni-directional chain for the synchronization of networked chaotic systems

In this paper, we introduce the structure of uni-directional chain (UDC) as a benchmark to probe the synchronizability of networked chaotic system. By studying the relation between the length of UDC and its synchronizability, we identify a region of coupling strength, referred to as strong controllable region (SCR), where a UDC with arbitrary length could realize synchronization that is robust to non-local disturbance. More importantly, it is found that there exists a kind of chaotic oscillators where the region of negative master stability function (MSF) overlaps with the SCR. As MSF implements local stability analysis to provide the necessary condition for the appearance of synchronization, while SCR permits stable synchronization corresponding to non-local disturbance pertinent to sufficient condition, the role of MSF for such kind of oscillator may be equivalent to provide sufficient condition of network synchronization. Besides, we propose methods for the control of network synchronization based on the characteristic of UDC. For networks with fixed structure, by obtaining a minimal spanning tree, each node chooses one of its neighbors and follows its dynamics with uniform coupling strength to realize synchronization. For growing networks, by establishing directed connection from newly added nodes to existing nodes. An interacting structure which is similar to the combination of a group of UDCs is formed and the system is guaranteed to be controlled within the SCR.