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- ItemPartially coherent twisted states in arrays of coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Omel'chenko, Oleh; Wolfrum, Matthias; Laing, CarloWe consider a one-dimensional array of phase oscillators with non-local coupling and a Lorenztian distribution of natural frequencies. The primary objects of interest are partially coherent states that are uniformly twisted in space. To analyze these we take the continuum limit, perform an Ott/Antonsen reduction, integrate over the natural frequencies and study the resulting spatio-temporal system on an unbounded domain. We show that these twisted states and their stability can be calculated explicitly. We find that stable twisted states with different wave numbers appear for increasing coupling strength in the wellknown Eckhaus scenario. Simulations of finite arrays of oscillators show good agreement with results of the analysis of the infinite system.
- ItemControlling unstable chaos: Stabilizing chimera states by feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Sieber, Jan; Omel'chenko, Oleh; Wolfrum, MatthiasWe present a control scheme that is able to find and stabilize a chaotic saddle in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is non-invasive, however, only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.
- ItemSpectral properties of chimera states(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Wolfrum, Matthias; Omel'chenko, Oleh; Yanchuk, Serhiy; Maistrenkko, YuriLiteraturverz. Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.
- ItemCoherence-incoherence patterns in a ring of non-locally coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Omel'chenko, OlehWe consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherence-incoherence patterns and to suggest directions for searching new ones
- ItemNonuniversal transitions to synchrony in the Sakaguchi-Kuramoto model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Omel'chenko, Oleh; Wolfrum, MatthiasWe investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (Sakaguchi-Kuramoto model). We show that for certain unimodal frequency distributions there appear unusual types of synchronization transitions, where synchrony can decay with increasing coupling, incoherence can regain stability for increasing coupling, or multistability between partially synchronized states and/or the incoherent state can appear. Our method is a bifurcation analysis based on a frequency dependent version of the Ott-Antonsen method and allows for a universal description of possible synchronization transition scenarios for any given distribution of natural frequencies. ...
- ItemStationary patterns of coherence and incoherence in two-dimensional arrays of non-locally coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Omel'chenko, Oleh; Wolfrum, Matthias; Yanchuk, Serhiy; Maistrenko, Yuri; Sudakov, OleksandrRecently it has been shown that large arrays of identical oscillators with non-local coupling can have a remarkable type of solutions that display a stationary macroscopic pattern of coexisting regions with coherent and incoherent motion, often called chimera states. We present here a detailed numerical study of the appearance of such solutions in twodimensional arrays of coupled phase oscillators. We discover a variety of stationary patterns, including circular spots, stripe patterns, and patterns of multiple spirals. Here, the stationarity means that for increasing system size the locally averaged phase distributions tend to the stationary profile given by the corresponding thermodynamic limit equation
- ItemBifurcations in the Sakaguchi-Kuramoto model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Omel'chenko, Oleh; Wolfrum, MatthiasWe analyze the Sakaguchi-Kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the Ott-Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non-universal synchronization transitions.
- ItemExistence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Omel'chenko, Oleh; Recke, LutzThis paper concerns general singularly perturbed second order semilinear elliptic equations on bounded domains $Omega subset R^n$ with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, we prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and we estimate the distance between the approximate and the exact solutions. Here ''spike solution'' means that there exists a point in $Omega$ such that the solution has a spike-like shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign.
- ItemRegular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Wolfrum, Matthias; Omel'chenko, Oleh; Sieber, JanWe study a system of phase oscillators with non-local coupling in a ring that supports self-organized patterns of coherence and incoherence, called chimera states. Introducing a global feedback loop, connecting the phase lag to the order parameter, we can observe chimera states also for systems with a small number of oscillators. Numerical simulations show a huge variety of regular and irregular patterns composed of localized phase slipping events of single oscillators. Using methods of classical finite dimensional chaos and bifurcation theory, we can identify the emergence of chaotic chimera states as a result of transitions to chaos via period doubling cascades, torus breakup, and intermittency. We can explain the observed phenomena by a mechanism of self-modulated excitability in a discrete excitable medium.