The field of network synchronization has seen tremendous growth following the introduction of the master stability function (MSF) formalism, which enables the efficient stability analysis of synchronization in large oscillator networks. However, to make further progress we must overcome the limitations of this celebrated formalism, which focuses on global synchronization and requires both the oscillators and their interaction functions to be identical, while many systems of interest are inherently heterogeneous and exhibit complex synchronization patterns. Here, we establish a generalization of the MSF formalism that can characterize the stability of any cluster synchronization pattern, even when the oscillators and/or their interaction functions are nonidentical. The new framework is based on finding the finest simultaneous block diagonalization of matrices in the variational equation and does not rely on information about network symmetry. This leads to an algorithm that is error-tolerant and orders of magnitude faster than existing symmetry-based algorithms. As an application, we rigorously characterize the stability of chimera states in networks with multiple types of interactions.
We study the synchronization and stability of power grids within the Kuramoto phase oscillator model with inertia with a bimodal frequency distribution representing the generators and the loads. We identify critical nodes through solitary frequency deviations and Lyapunov vectors corresponding to unstable Lyapunov exponents. To cure dangerous deviations from synchronization we propose time-delayed feedback control, which is an efficient control concept in nonlinear dynamic systems. Different control strategies are tested and compared with respect to the minimum number of controlled nodes requi…
When describing complex interconnected systems, one often has to go beyond the standard network description to account for generalized interactions. Here, we establish a unified framework to simplify the stability analysis of cluster synchronization patterns for a wide range of generalized networks, including hypergraphs, multilayer networks, and temporal networks. The framework is based on finding a simultaneous block diagonalization of the matrices encoding the synchronization pattern and the network topology. As an application, we use simultaneous block diagonalization to unveil an intrigui…
We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory all…
We propose a method for controlling synchronization patterns of limit-cycle oscillators by common noisy inputs, i.e., by utilizing noise-induced synchronization. Various synchronization patterns, including fully synchronized and clustered states, can be realized by using linear filters that generate appropriate common noisy signals from given noise. The optimal linear filter can be determined from the linear phase response property of the oscillators and the power spectrum of the given noise. The validity of the proposed method is confirmed by numerical simulations.
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 presence of general c…
Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is believed to enhance the stability of identical synchronization. Yet, here we show that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry. This counterintuitive effect holds for generic node dynamics and arbitrary network structure and is, moreover, robust against no…
Spectral properties of Coupled Map Lattices are described. Conditions for the stability of spatially homogeneous chaotic solutions are derived using linear stability analysis. Global stability analysis results are also presented. The analytical results are supplemented with numerical examples. The quadratic map is used for the site dynamics with different coupling schemes such as global coupling, nearest neighbor coupling, intermediate range coupling, random coupling, small world coupling and scale free coupling.
We investigate the stability of the synchronization manifold in a ring and an open-ended chain of nearest neighbors coupled self-sustained systems, each self-sustained system consisting of multi-limit cycles van der Pol oscillators. Such model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in brain waves model. The ring and open-ended chain of identical and non-identical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the com…
We study experimentally the synchronization patterns in time-delayed directed Boolean networks of excitable systems. We observe a transition in the network dynamics when the refractory time of the individual systems is adjusted. When the refractory time is on the same order-of-magnitude as the mean link time delays or the heterogeneities of the link time delays, cluster synchronization patterns change, or are suppressed entirely, respectively. We also show that these transitions occur when we only change the properties of a small number of nodes identified by their larger in-degree, hence the …
We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients $α_i$ that represent how strongly node $i$ disturbs the synchronization. With this approach, we can quantify the effect of individ…
We consider two coupled phase oscillators in the presence of proportional ("common") and independent white noises. The global synchronization properties of the system are analytically studied via the Fokker-Planck equation. When the "effective coupling" is big compared to the common noises, the former favors and the latter hinder synchronization. On the contrary, when the coupling is small compared to the proportional noises, we find that the latter induce synchronization, optimally when their intensities are big and in the n:m synchronization ratio. Furthermore, in such case a small value of …
In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that instead of microscopic details of the individual nodes/vertices rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitra…
The problem of synchronization in heterogeneous networks of linear systems with nonlinear delayed diffusive coupling is considered. The network is presented in new coordinates mean-field dynamics and synchronization errors. Thus the problem of network synchronization is reduced to the studying of synchronization-error system stability. The circle criterion for time-delay systems is used to derive the stability conditions of synchronization-error system. Obtained results are applied to a network of neural mass model populations, and the synchronization conditions are established. Simulation res…
Linear stability of synchronized states in networks of delay-coupled oscillators depends on the type of interaction, the network and oscillator properties. For inert oscillator response, found ubiquitously from biology to engineering, states with time-dependent frequencies can arise. These generate side bands in the frequency spectrum or lead to chaotic dynamics. Stability analysis is difficult due to delay-induced multistability and has only been available via numerical approaches. We derive criteria and conditions that enable fast and robust analytical linear stability analysis based on the …
We consider optimization of linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on overall coupling intensity and on stationary phase difference is derived. The improvement in the linear stability is illustrated by using several types of li…
A major challenge in neuroscience is posed by the need for relating the emerging dynamical features of brain activity with the underlying modular structure of neural connections, hierarchically organized throughout several scales. The spontaneous emergence of coherence and synchronization across such scales is crucial to neural function, while its anomalies often relate to pathological conditions. Here we provide a numerical study of synchronization dynamics in the human connectome network. Our purpose is to provide a detailed characterization of the recently uncovered broad dynamic regime, in…
We study synchronization patterns in repulsively coupled Kuramoto oscillators and focus on the impact of disorder in the natural frequencies. Among other choices we select the grid size and topology in a way that we observe a dynamically induced dimensional reduction with a continuum of attractors as long as the natural frequencies are uniformly chosen. When we introduce disorder in these frequencies, we find limit cycles with periods that are orders of magnitude longer than the natural frequencies of individual oscillators. Moreover we identify sequences of temporary patterns of phase-locked …
Bar-Ilan University（巴伊兰大学）；Indian Institute of Information Technology Raichur（赖布尔印度信息技术学院）；University of Petroleum and Energy Studies（石油与能源研究大学）；Indian Institute of Science Education and Research Thiruvananthapuram（蒂鲁文南特布勒姆印度科学教育与研究学院）
In this paper, synchronization of fractional order Coullet system with precise and also unknown parameters are studied. The proposed method which is based on the adaptive backstepping, has been developed to synchronize two chaotic systems with the same or partially different attractor. Sufficient conditions for the synchronization are analytically obtained. There after an adaptive control law is derived to make the states of two slightly mismatched chaotic Coullet systems synchronized. The stability analysis is then proved using the Lyapunov stability theorem. It is the privilege of the approa…
We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators (Okuda and Kuramoto, 1991). It includes three types of bistabilities, higher order entrainment and the existence of states with unusual stability properties. All possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically.
We analyze the physical mechanisms leading either to synchronization or to the formation of spatio-temporal patterns in a lattice model of pulse-coupled oscillators. In order to make the system tractable from a mathematical point of view we study a one-dimensional ring with unidirectional coupling. In such a situation, exact results concerning the stability of the fixed of the dynamic evolution of the lattice can be obtained. Furthermore, we show that this stability is the responsible for the different behaviors.
Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are linearly coupled. The nonlinear optimal interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in…
We study the interplay between network topology and complex space-time patterns and introduce a concept to analytically predict complex patterns in networks of Stuart-Landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology. These patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes. The underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks. In addition it directly links the sta…
We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict …
In the light of the recently proposed scenario of asymmetry-induced synchronization (AISync), in which dynamical uniformity and consensus in a distributed system would demand certain asymmetries in the underlying network, we investigate here the influence of some regularities in the interlayer connection patterns on the synchronization properties of multilayer random networks. More specifically, by considering a Stuart-Landau model of complex oscillators with random frequencies, we report for multilayer networks a dynamical behavior that could be also classified as a manifestation of AISync. W…
Learning Dynamic Stability Landscapes in Synchronization Networks
Department of Complexity Science, Postdam Institute for Climate Impact Research, Potsdam, Germany（复杂科学系，波茨坦气候影响研究所，德国波茨坦）；Department of Digital Transformation in Energy Systems, Institute of Energy Technology, Technical University of Berlin, Germany（能源系统数字化转型系，能源技术研究所，柏林技术大学，德国）；Machine Learning Group, Technical University of Berlin, 10587 Berlin, Germany（机器学习组，柏林技术大学，柏林，德国）
The production process of integrated electronic circuitry inherently leads to large heterogeneities on the component level. For electronic clock networks this implies detuned intrinsic frequencies and differences in coupling strength and the characteristic time-delays associated with signal transmission, processing and feedback. Using a phase-model description, we study the effects of such component heterogeneity on the dynamical properties of synchronization in networks of mutually delay-coupled Kuramoto oscillators. We test the theory against experimental results and circuit-level simulation…
Triadic subgraph analysis reveals the structural features in power networks based on higher-order connectivity patterns. Power networks have a unique triad significance profile (TSP) of the five unidirectional triadic subgraphs in comparison with the scale-free, small-world and random networks. Notably, the triadic closure has the highest significance in power networks. Thus, the unique TSP can serve as a structural identifier to differentiate power networks from other complex networks. Power networks form a network superfamily. Furthermore, synthetic power networks based on the random growth …
We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart--Landau oscillators. To this end a network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio synchronization tends to be unstable. However, increasing distributional widths can counteract this trend leading to multiple resynchronization transitions at relatively high inhibition ratios. All studies are performed on two distribution …
In real world, there is a significant relation between human behaviors and epidemic spread. Especially, the reactions among individuals in different communities to epidemics may be different, which lead to cluster synchronization of human behaviors. So, a mathematical model that embeds community structures, behavioral evolution and epidemic transmission is constructed to study the interaction between cluster synchronization and epidemic spread. The epidemic threshold of the model is obtained by using Gersgorin Lemma and dynamical system theory. By applying the Lyapunov stability method, the st…
Swarlamators are particles capable of synchronize and swarm. Here we study the effects produced by an external periodic stimulus over a system of swarmalators that move in two dimensions. When the particles are fixed and interact with equal strength (Kuramoto oscillators) their phases tend to synchronize and lock to the external stimulus if its intensity is sufficiently large. Here we show that in a system of swarmalators the force also shifts the phases and angular velocities leading to synchronization with the external frequency. However, the correlation between phase and spatial location de…
Complex multiplex networks consist of several subnetwork layers, which interact via pairwise inter-layer connections. Relay synchronization between distant layers which are not directly connected, but only via a relay layer, can be observed in multiplex networks. We study three-layer networks of discrete logistic maps, where each individual layer is a nonlocally coupled ring, and demonstrate scenarios of relay synchronization of complex patterns in the outer layers which interact via an intermediate layer. We find regimes of relay synchronization for chimera states, i.e., patterns of coexistin…
The features of animal population dynamics, for instance, flocking and migration, are often synchronized for survival under large-scale climate change or perceived threats. These coherent phenomena have been explained using synchronization models. However, such models do not take into account asynchronous and adaptive updating of an individual's status at each time. Here, we modify the Kuramoto model slightly by classifying oscillators as leaders or followers, according to their angular velocity at each time, where individuals interact asymmetrically according to their leader/follower status. …
We analyze partial synchronization patterns in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report a dynamical asymmetry between the hemispheres, induced by the natural structural asymmetry. We show that the dynamical asymmetry can be enhanced by introducing the inter-hemispheric coupling strength as a control parameter for partial synchronization patterns. We specify the possible modalities for existence of unihemispheric sleep in human brain, where one hemisphere sleeps while the other remains awake. In fact, this stat…
Nonlocally coupled oscillators with a phase lag self-organize into various patterns such as global synchronization, the twisted state, and the chimera state. In this paper, we consider nonlocally coupled oscillators that move on a ring by randomly exchanging their positions with the neighbors, and investigate the combined effects of phase lag and mobility on the collective phase dynamics. Spanning the whole range of phase lag and mobility, we show that mobility promotes synchronization for an attractive coupling, while it destroys coherence for a repulsive coupling. The transition behaviors ar…
