Dynamics of the Phase Oscillators with Plastic Couplings

Кasatkin, D. V.; Nekorkin, Vladimir I.

doi:10.1007/s11141-016-9662-1

Cited by 25 publications

(17 citation statements)

References 14 publications

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“…5(b), one-cluster states of antipodal type are supported by a Hebbian-like adaption (β ≈ −π/2) while splay states are supported by causal rules (β ≈ 0). For the asynchronous region, the dynamical system (1)- (2) can exhibit very complex dynamics and show chaotic motion 13 . This region is supported by an anti-Hebbian-like rule (β ≈ π/2).…”

Section: Stability Of One-cluster Statesmentioning

confidence: 99%

Hierarchical frequency clusters in adaptive networks of phase oscillators

Berner

Fialkowski

Кasatkin

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit.

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Section: Stability Of One-cluster Statesmentioning

confidence: 99%

Hierarchical frequency clusters in adaptive networks of phase oscillators

Berner

Fialkowski

Кasatkin

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Various synchronization patterns are known, like cluster synchronization where the network splits into groups of synchronous elements [14], or partial synchronization patterns like chimera states where the system splits into coexisting domains of coherent (synchronized) and incoherent (desynchronized) states [15][16][17]. These patterns were also explored in adaptive networks [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Furthermore, adapting the network topology has also successfully been used to control cluster synchronization in delay-coupled networks [34].Another focus of recent research in network science are multilayer networks, which are systems interconnected through different types of links [35][36][37][38].…”

mentioning

confidence: 99%

“…Eq. (S1) has been widely used as a paradigmatic model for adaptive networks [18][19][20][21][22][23][24][25][26][27][28][29][30]. It generalizes the Kuramoto-Sakaguchi model with fixed coupling topology [90][91][92][93][94].…”

mentioning

confidence: 99%

“…For a single layer, Eq. (S1) has been studied numerically and analytically [18][19][20][21][22]24]. In particular, it was shown that starting from uniformly distributed random initial condition φ i ∈ [0, 2π), κ ij ∈ [−1, 1] the system can reach different frequency multi-cluster states with hierarchical structure depending on the parameters α and β.…”

mentioning

confidence: 99%

“…It is known that already the monoplex system (S1) possesses a huge variety of dynamical states such as multi-clusters with respect to frequency synchronization, chaotic attractors, and chimera-like states [20][21][22][23][24]. In this article we will focus on the simplest patterns, namely one-cluster states, and their generalization to the multiplex case.…”

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Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks

2020

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We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuro-and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multi-layer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.Complex networks are an ubiquitous paradigm in nature and technology, with a wide field of applications ranging from physics, chemistry, biology, neuroscience, to engineering and socio-economic systems. Of particular interest are adaptive networks, where the connectivity changes in time, for instance, the synaptic connections between neurons are adapted depending on the relative timing of neuronal spiking [1][2][3][4][5]. Similarly, chemical systems have been reported [6], where the reaction rates adapt dynamically depending on the variables of the system. Activity-dependent plasticity is also common in epidemics [7] and in biological or social systems [8]. Synchronization is an important feature of the dynamics in networks of coupled nonlinear oscillators [9][10][11][12][13]. Various synchronization patterns are known, like cluster synchronization where the network splits into groups of synchronous elements [14], or partial synchronization patterns like chimera states where the system splits into coexisting domains of coherent (synchronized) and incoherent (desynchronized) states [15][16][17]. These patterns were also explored in adaptive networks [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Furthermore, adapting the network topology has also successfully been used to control cluster synchronization in delay-coupled networks [34].Another focus of recent research in network science are multilayer networks, which are systems interconnected through different types of links [35][36][37][38]. A prominent example are social networks which can be described as groups of people with different patterns of contacts or interactions between them [39][40][41]. Other applications are communication, supply, and transportation networks, for instance power grids, subway networks, or airtraffic networks [42]. In neuroscience, multilayer networks represent for instance neurons in different areas of the brain, neurons connected either by a chemical link or by an electrical synapsis, or...

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Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Berner¹

2021

Springer Theses

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Collective phenomena in systems of interconnected dynamical units are omnipresent in nature. The swarm behavior in flocks of birds or schools of fish, the synchronous flashing of fireflies or even coherent spiking of neurons in the human brain are just a few examples of collective motion. Elucidating the mechanisms that give rise to synchronization is crucial in order to understand biological self-organization. For this sake, the theory of dynamical networks has been successfully applied over the last decades to boil down the complex dynamics from natural systems to their essentials. In addition, dynamical networks with adaptive couplings and hence non-constant network structure appear naturally in real-world systems such as power grid networks, social networks as well as neuronal networks. In this thesis, we study adaptive networks and their properties which give rise to the emergence of a variety of synchronization patterns, including complete and cluster synchronization as well as solitary and chimera-like states. One of the main fields of application that is investigated in this thesis concerns neuroscience. However, the methods and approaches developed in this work are not restricted to a specific field of application.aus gekoppelten Phasenoszillatoren an und zeigen, wie das subtile Zusammenspiel zwischen Adaptivität und Netzwerkstruktur die Entstehung komplexer partieller Synchronisationsmuster hervorruft.Trotz des regen Interesses an adaptiven Netzwerken ist über das Zusammenspiel adaptiv gekoppelter Netzwerkgruppen wenig bekannt. Solche adaptiven Mehrschicht-oder Multiplex-Netzwerke kommen in neuronalen Netzwerken ganz natürlich vor. Wir schlagen ein Konzept zur Erzeugung und Stabilisierung diverser partieller Synchronisationsmuster (Phasen-Cluster) in adaptiven Netzwerken vor. Wir zeigen, dass das "Multiplexen" verschiedene stabile Phasen-Cluster-Zustände induzieren kann, selbst wenn diese in den Einschicht-Systemen nicht stabil sind oder gar nicht existieren. Weiterhin entwickeln wir eine Methode zur Analyse von Laplace-Matrizen von Multiplex-Netzwerken, die einen Einblick in die spektrale Struktur dieser Netzwerke ermöglicht und damit eine Reduktion des Stabilitätsproblems auf die von Einschicht-Netzwerken ermöglicht. Wir setzen die neue Methode ein, um analytische Ergebnisse für die Stabilität der Zustände im Mehrschicht-System zu erhalten.

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Dynamics of the Phase Oscillators with Plastic Couplings

Cited by 25 publications

References 14 publications

Hierarchical frequency clusters in adaptive networks of phase oscillators

Hierarchical frequency clusters in adaptive networks of phase oscillators

Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

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