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![Classes in the Master Stability Function
The maximum Lyapunov exponent Λ as a function of the parameter ν (see text for definitions), for a chaotic flow (Λ(0) > 0). The curve Λ(ν) is called the Master Stability Function (MSF). Given any pair of f and g, only three classes of systems are possible: Class I systems (yellow line) for which the MSF does not intercept the horizontal axis; Class II systems (violet line), for which the MSF has a unique intercept with the horizontal axis at ν = ν*; Class III systems (brown line), for which the MSF intercepts the horizontal axis at two critical points ν=ν1*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{1}^{*}$$\end{document} and ν=ν2*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{2}^{*}$$\end{document}.](publication/381314454/figure/fig1/AS:11431281251369655@1718207616535/Classes-in-the-Master-Stability-Function-The-maximum-Lyapunov-exponent-L-as-a-function-of.png)
Classes in the Master Stability Function The maximum Lyapunov exponent Λ as a function of the parameter ν (see text for definitions), for a chaotic flow (Λ(0) > 0). The curve Λ(ν) is called the Master Stability Function (MSF). Given any pair of f and g, only three classes of systems are possible: Class I systems (yellow line) for which the MSF does not intercept the horizontal axis; Class II systems (violet line), for which the MSF has a unique intercept with the horizontal axis at ν = ν*; Class III systems (brown line), for which the MSF intercepts the horizontal axis at two critical points ν=ν1*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{1}^{*}$$\end{document} and ν=ν2*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{2}^{*}$$\end{document}.
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We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which c...
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We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which c...
Citations
... However, cluster synchronization can clearly be discerned via the distribution of phases for these large lattices. We note that such effects have been observed in other systems and analyzed via eigenvalue analysis [45]. ...
The analysis of the synchronization of oscillator systems based on simplicial complexes presents some interesting features. The transition to synchronization can be abrupt or smooth depending on the substrate, the frequency distribution of the oscillators and the initial distribution of the phase angles. Both partial and complete synchronization can be seen as quantified by the order parameter. The addition of interactions of a higher order than the usual pairwise ones can modify these features further, especially when the interactions tend to have the opposite signs. Cluster synchronization is seen on sparse lattices and depends on the spectral dimension and whether the networks are mixed, sparse or compact. Topological effects and the geometry of shared faces are important and affect the synchronization patterns. We identify and analyze factors, such as frustration, that lead to these effects. We note that these features can be observed in realistic systems such as nanomaterials and the brain connectome.
... 2 ⟩ T , for the two clusters C 1 and C 2 , i.e., h = {1, 2} vs. the coupling strength d. The critical values predicted by the approach described in Ref. [21] are also reported in Fig. 3(b) as blue (orange) triangles for C 1 (C 2 ). ...
... A spectral block S localized at nodes {i 1 , i 2 , . . . , i N ′ } is defined as a subset of (N ′ − 1) eigenvectors of the Laplacian matrix L having the following properties [21]: ...
... As shown in [21], a group of nodes C is associated with a spectral block S if and only if they are equally connected (i.e., with the same weight) to each other node that is not in C, i.e. a ik = a jk , ∀i, j ∈ C and ∀k C. ...
Synchronization is a widespread phenomenon observed across natural and artificial networked systems. It often manifests itself by clusters of units exhibiting coincident dynamics. These clusters are a direct consequence of the organization of the Laplacian matrix eigenvalues into spectral localized blocks. We show how the concept of spectral blocks can be leveraged to design straightforward yet powerful controllers able to fully manipulate cluster synchronization of a generic network, thus shaping at will its parallel functioning. Specifically, we demonstrate how to induce the formation of spectral blocks in networks where such structures would not exist, and how to achieve precise mastering over the synchronizability of individual clusters by dictating the sequence in which each of them enters or exits the synchronization stability region as the coupling strength varies. Our results underscore the pivotal role of cluster synchronization control in shaping the parallel operation of networked systems, thereby enhancing their efficiency and adaptability across diverse applications.
... Besides the critical point characterizing the onset of synchronization (mainly concerned for coupled non-identical phase oscillators) and the critical point for global synchronization (mainly concerned for coupled identical chaotic oscillators), attention has been also paid to the transition from desynchronization to global synchronization in coupled oscillators. [53][54][55][56][57][58][59][60] In exploring the synchronization transition, two important issues are (1) how to identify the partially coherent states appeared at the intermediate coupling strengths and (2) how the system dynamics is transmitted among the partially coherent states Chaos ARTICLE pubs.aip.org/aip/cha as the coupling strength varies. Depending on the nodal dynamics and the network structure, the transition might present very different forms. ...
... This feature of non-smooth synchronization transition is more distinct for coupled chaotic oscillators, as the formation of synchronization clusters is dependent on the mesoscale network symmetries, which are discrete and rare for complex networks. [57][58][59][60] In some special cases, it is even shown that the increase in the coupling strength might deteriorate the synchronization degree of networked chaotic oscillators, i.e., a crossover is appeared in the transition. 58 By contrast, the transition reported in our present work is smooth at the oscillator level and, more interestingly, does not present a clear transition point. ...
Cluster synchronization in synthetic networks of coupled chaotic oscillators is investigated. It is found that despite the asymmetric nature of the network structure, a subset of the oscillators can be synchronized as a cluster while the other oscillators remain desynchronized. Interestingly, with the increase in the coupling strength, the cluster is expanding gradually by recruiting the desynchronized oscillators one by one. This new synchronization phenomenon, which is named “scalable synchronization cluster,” is explored theoretically by the method of eigenvector-based analysis, and it is revealed that the scalability of the cluster is attributed to the unique feature of the eigenvectors of the network coupling matrix. The transient dynamics of the cluster in response to random perturbations are also studied, and it is shown that in restoring to the synchronization state, oscillators inside the cluster are stabilized in sequence, illustrating again the hierarchy of the oscillators. The findings shed new light on the collective behaviors of networked chaotic oscillators and are helpful for the design of real-world networks where scalable synchronization clusters are concerned.
... When such cluster synchronization results from a graph symmetry, recent progress allows one to determine the onset, membership, and stability of the clusters of synchronized nodes [14][15][16]. However, synchronized clusters are prominent in networks such as neuron networks with connectivity structures that lack symmetries [17,18]. ...
Many real-world complex systems rely on cluster synchronization to function properly. A cluster of nodes exhibits synchronous behavior, while others behave erratically. Predicting the emergence of these clusters and understanding the mechanism behind their structure and variation in response to a parameter change is a daunting task in networks that lack symmetry. We unravel the mechanism for the emergence of cluster synchronization in heterogeneous random networks. We develop heterogeneous mean-field approximation together with a self-consistent theory to determine the onset and stability of the cluster. Our analysis shows that cluster synchronization occurs in a wide variety of heterogeneous networks, node dynamics, and coupling functions. The results could lead to a new understanding of the dynamical behavior of networks ranging from neural to social.
Synchronous firing of neurons is a key mechanism for ensuring coordination between neurons and effective
integration of information. Neuronal networks change over time and are influenced by external stimuli, and
exploring how these factors affect the synchronization capabilities of the network and their relationship with
energy fluctuations is an important issue. This study investigates the synchronization and energy changes of a
memristive FitzHugh-Nagumo neuronal network with time-varying topology under external stimuli. The results
show that external stimuli affect the synchronization of the network. Larger coupling strength, along with
appropriate amplitude and frequency, can effectively drive the network from an asynchronous to a synchronous
state. Furthermore, as neurons tend to synchronize, their energy changes become increasingly consistent. The
conditions for stable synchronization of the network are analyzed using the master stability function method,
confirming the consistency between theoretical and numerical results and revealing the key role of rewiring
probability in synchronization. The generality of the results is confirmed by varying network size and average
degree. These findings indicate that external stimuli and time-varying topology play crucial roles in regulating
collective behaviors in neuronal networks, providing new perspectives and theoretical support for the functional
understanding and regulation of the nervous system.
