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Author: Omelchenko, Oleh E. × End date: 2017 × Start date: 2017 × WGL Institute: WIAS ×

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Now showing 1 - 5 of 5
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    Phase sensitive excitability of a limit cycle
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Franovic, Igor; Omelchenko, Oleh E.; Wolfrum, Matthias
    The classical notion of excitability refers to an equilibrium state that shows under the influence of perturbations a nonlinear threshold-like behavior. Here, we extend this concept by demonstrating how periodic orbits can exhibit a specific form of excitable behavior where the nonlinear threshold-like response appears only after perturbations applied within a certain part of the periodic orbit, i.e the excitability happens to be phase sensitive. As a paradigmatic example of this concept we employ the classical FitzHugh-Nagumo system. The relaxation oscillations, appearing in the oscillatory regime of this system, turn out to exhibit a phase sensitive nonlinear thresholdlike response to perturbations, which can be explained by the nonlinear behavior in the vicinity of the canard trajectory. Triggering the phase sensitive excitability of the relaxation oscillations by noise we find a characteristic non-monotone dependence of the mean spiking rate of the relaxation oscillation on the noise level. We explain this non-monotone dependence as a result of an interplay of two competing effects of the increasing noise: the growing efficiency of the excitation and the degradation of the nonlinear response.
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    Surfing the edge: Finding nonlinear solutions using feedback control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Willis, Ashley P.; Duguet, Yohann; Omelchenko, Oleh E.; Wolfrum, Matthias
    Many transitional wall-bounded shear flows are characterised by the coexistence in statespace of laminar and turbulent regimes. Probing the edge boundary between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier-Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Travelling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.
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    Optimal design of the tweezer control for chimera states
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Omelchenko, Iryna; Omelchenko, Oleh E.; Zakharova, Anna; Schöll, Eckehard
    Chimera states are complex spatio-temporal patterns, which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyse the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.
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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.