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Author: Omelchenko, Oleh E. × Subject: Coupled oscillators ×

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Now showing 1 - 3 of 3
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    Turbulence in the Ott-Antonsen equation for arrays of coupled phase oscillators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wolfrum, Matthias; Gurevich, Svetlana V.; Omelchenko, Oleh E.
    In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott-Antonsen equation, which is an integrodifferential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it has been reported that in the spatially extended case at the synchronization threshold there appear partially coherent plane waves with different wave numbers, which are organized in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto-Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin-Feir type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves.We give an analytic treatment of the Benjamin-Feir instability and its onset in a codimension-two bifurcation in the Ott-Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin-Feir unstable region.
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    The mathematics behind chimera states
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Omelchenko, Oleh E.
    Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott-Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.
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    Stability of spiral chimera states on a torus
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Omelchenko, Oleh E.; Wolfrum, Matthias; Knobloch, Edgar
    We study destabilization mechanisms of spiral coherence-incoherence patterns known as spiral chimera states that form on a two-dimensional lattice of nonlocally coupled phase oscillators. For this purpose we employ the linearization of the OttAntonsen equation that is valid in the continuum limit and perform a detailed two-parameter stability analysis of a D4-symmetric chimera state, i.e., a four-core spiral state. We identify fold, Hopf and parity-breaking bifurcations as the main mechanisms whereby spiral chimeras can lose stability. Beyond these bifurcations we find new spatio-temporal patterns, in particular, quasiperiodic chimeras, D2-symmetric spiral chimeras as well as drifting states.