Recent advances in coupled oscillator theory

Ermentrout, Bard; Park, Youngmin; Wilson, Dan

doi:10.1098/rsta.2019.0092

Cited by 42 publications

(33 citation statements)

References 42 publications

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“…Recent studies, in addition, aim to make phase oscillators models even more powerful by generalizing the conditions under which the reduction techniques are valid [KLI17a,ROS19a,ROS19b,ERM19] As a representative from the class of phase oscillator models, the Kuramoto model where all oscillators are coupled in an all-to-all manner, particularly, has attracted a lot of attention due to its simple form and mathematical tractability [KUR84,STR00]. The Kuramoto model has gained additional popularity due to its application for real-world problems [STR93, PIK01, STR03, STR05a, ROD16].…”

Section: The Role Of Phase Oscillator Models For Complex Dynamical Networkmentioning

confidence: 99%

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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Section: The Role Of Phase Oscillator Models For Complex Dynamical Networkmentioning

confidence: 99%

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Berner¹

2021

Springer Theses

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“…Finally, before proceeding, we note that there are also other formulations of phase or phaseamplitude reduced equations for analyzing higher-order effects of perturbations on limit cycles described by ODEs, such as non-pairwise phase interactions [23], higher-order phase reduction [49], nonlinear phase coupling function [50], and higher-order approximations of coupling functions [41], which can capture more detailed aspects of synchronization than the lowest-order phase equation.…”

Section: Nonlinear Phase-amplitude Equationsmentioning

confidence: 99%

Nonlinear phase-amplitude reduction of delay-induced oscillations

Kotani

Ogawa

Shirasaka

et al. 2020

Phys. Rev. Research

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Spontaneous oscillations induced by time delays are observed in many real-world systems. Phase reduction theory for limit-cycle oscillators described by delay-differential equations (DDEs) has been developed to analyze their synchronization properties, but it is applicable only when the perturbation applied to the oscillator is sufficiently weak. In this study, we formulate a nonlinear phase-amplitude reduction theory for limit-cycle oscillators described by DDEs on the basis of the Floquet theorem, which is applicable when the oscillator is subjected to perturbations of moderate intensity. We propose a numerical method to evaluate the leading Floquet eigenvalues, eigenfunctions, and adjoint eigenfunctions necessary for the reduction and derive a set of low-dimensional nonlinear phase-amplitude equations approximately describing the oscillator dynamics. By analyzing an analytically tractable oscillator model with a cubic nonlinearity, we show that the asymptotic phase of the oscillator state in an infinite-dimensional state space can be approximately evaluated and non-trivial bistability of the oscillation amplitude caused by moderately strong periodic perturbations can be predicted on the basis of the derived phase-amplitude equations. We further analyze a model of generegulatory oscillator and illustrate that the reduced equations can elucidate the mechanism of its complex dynamics under non-weak perturbations, which may be relevant to real physiological phenomena such as circadian rhythm sleep disorders.

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“…In real systems, the form of equations (1.1) is obviously distorted (see [33][34][35], for example), and the generalization of the OA theory to non-ideal situations was a resisting challenge for a decade. A way out has been proposed recently in the form of circular cumulant approach [3,[36][37][38].…”

Section: (B) Opportunities Of the Ott-antonsen Theory And Its Generalizationmentioning

confidence: 99%

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott–Antonsen theory

Tyulkina

Goldobin

Klimenko

et al. 2020

Phil. Trans. R. Soc. A.

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The problem of magnetic transitions between the low-temperature (macrospin ordered) phases in two-dimensional XY arrays is addressed. The system is modelled as a plane structure of identical single-domain particles arranged in a square lattice and coupled by the magnetic dipole–dipole interaction; all the particles possess a strong easy-plane magnetic anisotropy. The basic state of the system in the considered temperature range is an antiferromagnetic (AF) stripe structure, where the macrospins (particle magnetic moments) are still involved in thermofluctuational motion: the superparamagnetic blocking T b temperature is lower than that ( T af ) of the AF transition. The description is based on the stochastic equations governing the dynamics of individual magnetic moments, where the interparticle interaction is added in the mean-field approximation. With the technique of a generalized Ott–Antonsen theory, the dynamics equations for the order parameters (including the macroscopic magnetization and the AF order parameter) and the partition function of the system are rigorously obtained and analysed. We show that inside the temperature interval of existence of the AF phase, a static external field tilted to the plane of the array is able to induce first-order phase transitions from AF to ferromagnetic state; the phase diagrams displaying stable and metastable regions of the system are presented. This article is part of the theme issue ‘Patterns in soft and biological matters’.

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Recent advances in coupled oscillator theory

Cited by 42 publications

References 42 publications

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Patterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators

Nonlinear phase-amplitude reduction of delay-induced oscillations

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott–Antonsen theory

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