Stable Chimeras and Independently Synchronizable Clusters

Abstract

Cluster synchronization is a phenomenon in which a network self-organizes into a pattern of synchronized sets. It has been shown that diverse patterns of stable cluster synchronization can be captured by symmetries of the network. Here we establish a theoretical basis to divide an arbitrary pattern of symmetry clusters into independently synchronizable cluster sets, in which the synchronization stability of the individual clusters in each set is decoupled from that in all the other sets. Using this framework, we suggest a new approach to find permanently stable chimera states by capturing two or more symmetry clusters-at least one stable and one unstable-that compose the entire fully symmetric network.Synchronization is a collective network behavior in which the states of the interacting units evolve in step with each other [1], as observed in animal flocking [2], the coordinated firing of neurons [3][4][5], and the synchronous operation of power generators [6]. Beyond the complete synchronization of all units, significant progress has been made on understanding more complex forms of synchronization, including cluster synchronization [7][8][9][10][11][12][13] and chimera states [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. In particular, cluster synchronization (CS), in which clusters of nodes exhibit synchronized dynamics, has seen recent breakthroughs: rigorous relations based on group theory have been established between patterns of synchronous clusters and the symmetries of the network structure [11,12]. Network symmetry can be used to explain various forms of collective behavior, such as remote synchronization, in which two nodes are synchronized despite being connected only through asynchronous ones [28], and isolated desynchronization, in which the synchronization of some clusters is broken without disturbing other clusters [11,12,29,30].Chimera states, which are characterized by the coexistence of both coherent and incoherent dynamics within the same state, are also intimately related to symmetry. Since the initial discovery [14] and subsequent analysis [15] of such states, numerous studies have found-numerically, analytically, and experimentally-that chimera states can be observed in a wide range of systems (see the review in Ref. [22] and the references therein). However, it was recently found that chimeras in finite-size networks can be long-lived but transient states [17] (i.e., the system will eventually settle onto a simpler state, such as complete synchronization). This raised a fundamental question: are permanently stable chimeras possible with a finite number of oscillators [31]? Evidence for the affirmative answer has so far been limited to numerical simulations [21,23,26] (notable exceptions are two case studies with stability analysis: one for "weak" chimeras in bistable populations of phase oscillators [27] and the other for a four-node network of delay-coupled opto-electronic oscillators [25]). Our approach for addressing this problem is to identify symmetry-based "templates" fo...

Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

“…Remark 1: (Network symmetries, equitable partitions, and balanced weights) Conditions to ensure the invariance of the cluster synchronization manifold have been linked to network symmetries, which are defined by the group comprising all node permutations that leave the network topology unchanged, e.g., see [31], [32], [44]. In Fig.…”

“…Invariance of exact cluster synchronization, which is the notion used in this paper, is also studied in [41], [42]. Stability of exact cluster synchronization is investigated in [43] where, however, only the restrictive case of two clusters for identical Kuramoto oscillators with inertia is considered, and in [44], where only implicit and numerical stability conditions are provided. To the best of our knowledge, our work presents the first explicit analytical conditions for the (local) stability of the cluster synchronization manifold in sparse and weighted networks of heterogeneous Kuramoto oscillators.…”

Critical Switching in Globally Attractive Chimeras

“…For finite-size systems, some chimera states have been shown to be long transients [38], while others have been shown to be stable [39,40] using the Watanabe-Strogatz ansatz [41,42]. Recent research has placed increased emphasis on chimeras in finite-size networks of chaotic oscillators [43][44][45][46][47], which are important given the prevalence of chaos in physical systems [48]. In that context, it has been shown that the stability of chimera states can be studied rigorously using cluster synchronization techniques [46,47].…”

“…Recent research has placed increased emphasis on chimeras in finite-size networks of chaotic oscillators [43][44][45][46][47], which are important given the prevalence of chaos in physical systems [48]. In that context, it has been shown that the stability of chimera states can be studied rigorously using cluster synchronization techniques [46,47].…”

Cluster synchronization in complex network of coupled chaotic circuits: An experimental study

“…Additional evidences from large-size complex network possessing permutation symmetries have been also provided [43][44][45][46][47][48]. With the help of computational group theory algorithm [50], the permutation symmetries of large-size complex networks now can be identified numerically, which, combining with the generalized method of eigenvalue analysis [44,46,49,51,52], can be used to analyze the formation of cluster synchronization in the general complex networks.…”