Filters

2013 - 20178

More filters

20198
Reset filters

Settings

Type: Report × Subject: Coupled oscillators ×

Search Results

Now showing 1 - 8 of 8
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Noise enhanced coupling between two oscillators with long-term plasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lücken, Leonhard; Popovych, Oleksandr V.; Tass, Peter A.; Yanchuk, Serhiy
    Spike time-dependent plasticity is a fundamental adaptation mechanism of the nervous system. It induces structural changes of synaptic connectivity by regulation of coupling strengths between individual cells depending on their spiking behavior. As a biophysical process its functioning is constantly subjected to natural fluctuations. We study theoretically the influence of noise on a microscopic level by considering only two coupled neurons. Adopting a phase description for the neurons we derive a two-dimensional system which describes the averaged dynamics of the coupling strengths. We show that a multistability of several coupling configurations is possible, where some configurations are not found in systems without noise. Intriguingly, it is possible that a strong bidirectional coupling, which is not present in the noise-free situation, can be stabilized by the noise. This means that increased noise, which is normally expected to desynchronize the neurons, can be the reason for an antagonistic response of the system, which organizes itself into a state of stronger coupling and counteracts the impact of noise. This mechanism, as well as a high potential for multistability, is also demonstrated numerically for a coupled pair of Hodgkin-Huxley neurons.
  • Thumbnail Image
    Item
    Amplitude equations for collective spatio-temporal dynamics in arrays of coupled systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Yanchuk, Serhiy; Perlikowski, Przemysław; Wolfrum, Matthias; Stefański, Andrzej; Kapitaniak, Tomasz
    We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Is there an impact of small phase lags in the Kuramoto model?
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Omelchenko, Oleh; Wolfrum, Matthias
    We discuss the influence of small phase lags on the synchronization transitions in the Kuramoto model for a large inhomogeneous population of globally coupled phase oscillators. Without a phase lag, all unimodal distributions of the natural frequencies give rise to a classical synchronization scenario, where above the onset of synchrony at the Kuramoto threshold there is an increasing synchrony for increasing coupling strength. We show that already for arbitrarily small phase lags there are certain unimodal distributions of natural frequencies such that for increasing coupling strength synchrony may decrease and even complete incoherence may regain stability. Moreover, our example allows a qualitative understanding of the mechanism for such non-universal synchronization transitions
  • Thumbnail Image
    Item
    Regular 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, Jan
    We 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.