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Author: Wolfrum, Matthias × Type: Text ×

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Now showing 1 - 10 of 40
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    Oscillatory instability in systems with delay
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Wolfrum, Matthias; Yanchuk, Serhiy
    Any biological or physical system, which incorporates delayed feedback or delayed coupling, can be modeled by a dynamical system with delayed argument. We describe a standard oscillatory destabilization mechanism, which occurs in such systems.
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    Partially coherent twisted states in arrays of coupled phase oscillators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Omel'chenko, Oleh; Wolfrum, Matthias; Laing, Carlo
    We 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.
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    Controlling unstable chaos: Stabilizing chimera states by feedback
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Sieber, Jan; Omel'chenko, Oleh; Wolfrum, Matthias
    We 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.
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    Noise-induced dynamical regimes in a system of globally coupled excitable units
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Klinshov, Vladimir V.; Kirillov, Sergey Yu.; Nekorkin, Vladimir I.; Wolfrum, Matthias
    We study the interplay of global attractive coupling and individual noise in a system of identical active rotators in the excitable regime. Performing a numerical bifurcation analysis of the nonlocal nonlinear Fokker-Planck equation for the thermodynamic limit, we identify a complex bifurcation scenario with regions of different dynamical regimes, including collective oscillations and coexistence of states with different levels of activity. In systems of finite size this leads to additional dynamical features, such as collective excitability of different types, noise-induced switching and bursting. Moreover, we show how characteristic quantities such as macroscopic and microscopic variability of inter spike intervals can depend in a non-monotonous way on the noise level.
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    Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Eydam, Sebastian; Franovic, Igor; Wolfrum, Matthias
    We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.
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    Control of unstable steady states by strongly delayed feedback
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Yanchuk, Serhiy; Wolfrum, Matthias; Hövel, Philipp; Schöll, Eckehard
    We present an asymptotic analysis of time-delayed feedback control of steady states for large delay time. By scaling arguments, and a detailed comparison with exact solutions, we establish the parameter ranges for successful stabilization of an unstable fixed point of focus type. Insight into the control mechanism is gained by analysing the eigenvalue spectrum, which consists of a pseudo-continuous spectrum and up to two strongly unstable eigenvalues. Although the standard control scheme generally fails for large delay, we find that if the uncontrolled system is sufficiently close to its instability threshold, control does work even for relatively large delay times.
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    Spectral properties of chimera states
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Wolfrum, Matthias; Omel'chenko, Oleh; Yanchuk, Serhiy; Maistrenkko, Yuri
    Literaturverz. 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.
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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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    Absolute stability and absolute hyperbolicity in systems with discrete time-delays
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Yanchuk, Serhiy; Wolfrum, Matthias; Pereira, Tiago; Turaev, Dmitry
    An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete timedelays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.
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    Chimera states are chaotic transients
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Wolfrum, Matthias; Omelʹčenko, Oleh
    Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states", has been described, where in a spatially homogeneous system regions of irregular incoherent motion coexist with regular synchronized motion, forming a self organized pattern in a population of nonlocally coupled oscillators. Whereas most of the previous studies of chimera states focused their attention to the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, we investigate here the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time-span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.