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- ItemSpectrum and amplitude equations for scalar delay-differential equations with large delay(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Yanchuk, Serhiy; Lücken, Leonhard; Wolfrum, Matthias; Mielke, AlexanderThe subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.
- ItemEmbedding the dynamics of a single delay system into a feed-forward ring(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Klinshov, Vladimir; Shchapin, Dmitry; Yanchuk, Serhiy; Wolfrum, Matthias; D'Huys, Otti; Nekorkin, VladimirWe investigate the relation between the dynamics of a single oscillator with delayed selffeedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where stability of periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHughNagumo type.
- 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. ...
- ItemIs there an impact of small phase lags in the Kuramoto model?(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Omelchenko, Oleh; Wolfrum, MatthiasWe 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
- ItemImproving the modulation bandwidth in semiconductor lasers by passive feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Radziunas, Mindaugas; Glitzky, Annegret; Bandelow, Uwe; Wolfrum, Matthias; Troppenz, Ute; Kreissl, Jochen; Rehbein, WolfgangWe explore the concept of passive-feedback lasers for direct signal modulation at 40 Gbit/s. Based on numerical simulation and bifurcation analysis, we explain the main mechanisms in these devices which are crucial for modulation at high speed. The predicted effects are demonstrated experimentally by means of correspondingly designed devices. In particular a significant improvement of the modulation bandwidth at low injection currents can be demonstrated.
- ItemAuditory cortex modelled as a dynamical network of oscillators: Understanding event-related fields and their adaptation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hajizadeh, Aida; Matysiak, Artur; Wolfrum, Matthias; May, Patrick J. C.Adaptation, the reduction of neuronal responses by repetitive stimulation, is a ubiquitous feature of auditory cortex (AC). It is not clear what causes adaptation, but short-term synaptic depression (STSD) is a potential candidate for the underlying mechanism. We examined this hypothesis via a computational model based on AC anatomy, which includes serially connected core, belt, and parabelt areas. The model replicates the event-related field (ERF) of the magnetoencephalogram as well as ERF adaptation. The model dynamics are described by excitatory and inhibitory state variables of cell populations, with the excitatory connections modulated by STSD. We analysed the system dynamics by linearizing the firing rates and solving the STSD equation using time-scale separation. This allows for characterization of AC dynamics as a superposition of damped harmonic oscillators, so-called normal modes. We show that repetition suppression of the N1m is due to a mixture of causes, with stimulus repetition modifying both the amplitudes and the frequencies of the normal modes. In this view, adaptation results from a complete reorganization of AC dynamics rather than a reduction of activity in discrete sources. Further, both the network structure and the balance between excitation and inhibition contribute significantly to the rate with which AC recovers from adaptation. This lifetime of adaptation is longer in the belt and parabelt than in the core area, despite the time constants of STSD being spatially constant. Finally, we critically evaluate the use of a single exponential function to describe recovery from adaptation.
- 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.
- ItemSlow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Wolfrum, Matthias; Ehrt, JuliaIn this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions.
- ItemExternal cavity modes in Lang-Kobayashi and traveling wave models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Radziunas, Mindaugas; Wünsche, Hans-Jürgen; Krauskopf, Bernd; Wolfrum, MatthiasWe investigate a semiconductor laser with delayed optical feedback due to an external cavity formed by a regular mirror. We discuss similarities and differences of the well-known Lang--Kobayashi delay differential equation model and the traveling wave partial differential equation model. For comparison we locate the continuous wave states in both models and analyze their stability.
- ItemAmplitude 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, TomaszWe 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.