Bifurcations and Singularities for Coupled Oscillators with Inertia and Frustration.
Julien Barré,David Métivier +1 more
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It is proved that any nonzero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous or from discontinuous to continuous.Abstract:
We prove that any nonzero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.read more
Citations
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A First Order Phase Transition Resulting from Finite Inertia in Coupled Oscillator Systems
TL;DR: In this article, the collective behavior of a set of coupled damped driven pendula with finite inertia was analyzed, and it was shown that the synchronization of the oscillators exhibits a first order phase transition synchronization onset, substantially different from the second order transition obtained in the case of no inertia.
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Statistical mechanics and dynamics of solvable models with long-range interactions
TL;DR: A comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions can be found in this article, where the core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles of mean-field type models.
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What adaptive neuronal networks teach us about power grids.
TL;DR: It is proved that phase oscillator models with inertia can be viewed as a particular class of adaptive networks, and the phenomenon of cascading line failure in power grids is translated into an adaptive neuronal network.
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Structure preserving schemes for the continuum Kuramoto model: Phase transitions
TL;DR: Numerical methods which are capable to preserve structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case are developed.
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Blinking chimeras in globally coupled rotators.
TL;DR: This work describes a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia) characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new reshuffled configuration.
References
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Book
Synchronization: A Universal Concept in Nonlinear Sciences
TL;DR: This work discusseschronization of complex dynamics by external forces, which involves synchronization of self-sustained oscillators and their phase, and its applications in oscillatory media and complex systems.
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The Kuramoto model: A simple paradigm for synchronization phenomena
TL;DR: In this paper, a review of the Kuramoto model of coupled phase oscillators is presented, with a rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years.
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From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators
TL;DR: In this article, the authors review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto's work to Crawford's recent contributions.
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Biological rhythms and the behavior of populations of coupled oscillators
TL;DR: It is proposed that self-entraining communities of this sort may exist within individual metazoan animals and plants as the basis of the observed diurnal coordination of their physiological process.
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Low dimensional behavior of large systems of globally coupled oscillators.
Edward Ott,Thomas M. Antonsen +1 more
TL;DR: It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics and an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered are derived.