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Author: Omelchenko, Oleh E. × Subject: feedback control ×

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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.