### Browsing by Author "Wolfrum, Matthias"

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- ItemAbsolute 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, DmitryAn 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.
- 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.
- 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.
- 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.
- ItemBumps, chimera states, and Turing patterns in systems of coupled active rotators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Franović, Igor; Omel'chenko, Oleh E.; Wolfrum, MatthiasSelf-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest, are called bump states. Here, we study bumps in an array of active rotators coupled by non-local attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.
- ItemChimera states are chaotic transients(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Wolfrum, Matthias; Omelʹčenko, OlehSpatiotemporal 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.
- ItemCoexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Burylko, Oleksandr; Mielke, Alexander; Wolfrum, Matthias; Yanchuk, SerhiyWe consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i.e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.
- ItemControl of unstable steady states by strongly delayed feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Yanchuk, Serhiy; Wolfrum, Matthias; Hövel, Philipp; Schöll, EckehardWe 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.
- 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.
- ItemDestabilization patterns in large regular networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Yanchuk, Serhiy; Wolfrum, MatthiasWe describe a generic mechanism for the destabilization in large regular networks of identical coupled oscillators. Based on a reduction method for the spectral problem, we first present a criterion for this type of destabilization. Then, we investigate the related bifurcation scenario, showing the existence of a large number of coexisting periodic solutions with different frequencies, spatial patterns, and stability properties. Even for unidirectional coupling this can be understood in analogy to the well-known Eckhaus scenario for diffusive systems.
- ItemDynamics of a stochastic excitable system with slowly adapting feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Franović, Igor; Yanchuk, Serhiy; Eydam, Sebastian; Bačić, Iva; Wolfrum, MatthiasWe study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker-Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.
- 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.
- 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.
- ItemGeometry of heteroclinic cascades in scalar parabolic differential equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 1998) Wolfrum, MatthiasWe investigate the geometrical properties of the attractor for semilinear scalar parabolic PDEs on a bounded interval with Neumann boundary conditions. Using the nodal properties of the stationary solutions which are determined by an ordinary boundary value problem, we obtain crucial information about the long-time behavior for the full PDE. Especially, we prove a criterion for the intersection of strong- stable and unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.
- 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.
- 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
- ItemLeap-frog patterns in systems of two coupled FitzHugh-Nagumo units(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Eydam, Sebastian; Franovic, Igor; Wolfrum, MatthiasWe 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.
- ItemThe link between coherence echoes and mode locking(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Eydam, Sebastian; Wolfrum, MatthiasWe investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally- coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self- organized periodic pulsations of the mean field, called mode locking, have been described re- cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu- ral frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as the result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimula- tion induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the mode-locking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The non-monotonous behavior of echo amplitudes can be explained as a result of the linear dispersion within the self-organized mode-locked frequency comb. Fi- nally we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus.
- ItemMode-locking in systems of globally coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Eydam, Sebastian; Wolfrum, MatthiasWe investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode-locking in lasers and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonic in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode-locking, transitions to phase turbulence and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type-II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence.
- ItemMultiple self-locking in the Kuramoto--Sakaguchi system with delay(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Wolfrum, Matthias; Yanchuk, Serhiy; D'Huys, OttiWe study the Kuramoto-Sakaguchi system of phase oscillators with a delayed mean-field coupling. By applying the theory of large delay to the corresponding Ott--Antonsen equation, we explain fully analytically the mechanisms for the appearance of multiple coexisting partially locked states. Closely above the onset of synchronization, these states emerge in the Eckhaus scenario: with increasing coupling, more and more partially locked states appear unstable from the incoherent state, and gain stability for larger coupling at a modulational stability boundary. The partially locked states with strongly detuned frequencies are shown to emerge subcritical and gain stability only after a fold and a series of Hopf bifurcations. We also discuss the role of the Sakaguchi phase lag parameter. For small delays, it determines, together with the delay time, the attraction or repulsion to the central frequency, which leads to supercritical or subcritical behavior, respectively. For large delay, the Sakaguchi parameter does not influence the global dynamical scenario.