Browsing by Author "Yanchuk, Serhiy"
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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.
- 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.
- ItemDelay-induced patterns in a two-dimensional lattice of coupled oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Kantner, Markus; Schöll, Eckehard; Yanchuk, SerhiyWe show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators. A "hybrid dispersion relation" is introduced, which describes the stability of the patterns in spatially extended systems with large time-delay.
- 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.
- ItemDetection and storage of multivariate temporal sequences by spiking pattern reverberators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lücken, Leonhard; Yanchuk, SerhiyWe consider networks of spiking coincidence detectors in continuous time. A single detector is a finite state machine that emits a pulsatile signal whenever the number incoming inputs exceeds a threshold within a time window of some tolerance width. Such finite state models are well-suited for hardware implementations of neural networks, as on integrated circuits (IC) or field programmable arrays (FPGAs) but they also reflect the natural capability of many neurons to act as coincidence detectors. We pay special attention to a recurrent coupling structure, where the delays are tuned to a specific pattern. Applying this pattern as an external input leads to a self-sustained reverberation of the encoded pattern if the tuning is chosen correctly. In terms of the coupling structure, the tolerance and the refractory time of the individual coincidence detectors, we determine conditions for the uniqueness of the sustained activity, i.e., for the funcionality of the network as an unambiguous pattern detector. We also present numerical experiments, where the functionality of the proposed pattern detector is demonstrated replacing the simplistic finite state models by more realistic Hodgkin-Huxley neurons and we consider the possibility of implementing several pattern detectors using a set of shared coincidence detectors. We propose that inhibitory connections may aid to increase the precision of the pattern discrimination.
- ItemDynamical phenomena in complex networks: fundamentals and applications(Berlin ; Heidelberg : Springer, 2021) Yanchuk, Serhiy; Roque, Antonio C.; Macau, Elbert E. N.; Kurths, JürgenThis special issue presents a series of 33 contributions in the area of dynamical networks and their applications. Part of the contributions is devoted to theoretical and methodological aspects of dynamical networks, such as collective dynamics of excitable systems, spreading processes, coarsening, synchronization, delayed interactions, and others. A particular focus is placed on applications to neuroscience and Earth science, especially functional climate networks. Among the highlights, various methods for dealing with noise and stochastic processes in neuroscience are presented. A method for constructing weighted networks with arbitrary topologies from a single dynamical node with delayed feedback is introduced. Also, a generalization of the concept of geodesic distances, a path-integral formulation of network-based measures is developed, which provides fundamental insights into the dynamics of disease transmission. The contributions from the Earth science application field substantiate predictive power of climate networks to study challenging Earth processes and phenomena.
- ItemDynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Yanchuk, Serhiy; Giacomelli, GiovanniDynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.
- 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.
- ItemEmergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Klinshov, Vladimir; Lücken, Leonhard; Shchapin, Dmitry; Nekorkin, Vladimir; Yanchuk, SerhiyInteraction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillators phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous jittering regimes with non-equal interspike intervals (ISIs). The number of the emergent solutions increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how each periodic solution consisting of different ISIs implies the appearance of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.
- ItemFrom synchronization to Lyapunov exponents and back(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Politi, Antonio; Ginelli, Francesco; Yanchuk, Serhiy; Maistrenko, YuriThe goal of this paper is twofold. In the first part we discuss a general approach to determine Lyapunov exponents from ensemble- rather than time-averages. The approach passes through the identification of locally stable and unstable manifolds (the Lyapunov vectors), thereby revealing an analogy with generalized synchronization. The method is then applied to a periodically forced chaotic oscillator to show that the modulus of the Lyapunov exponent associated to the phase dynamics increases quadratically with the coupling strength and it is therefore different from zero already below the onset of phase-synchronization. The analytical calculations are carried out for a model, the generalized special flow, that we construct as a simplified version of the periodically forced Rössler oscillator.
- ItemMaster Memory Function for Delay-Based Reservoir Computers With Single-Variable Dynamics([New York, NY] : IEEE, 2022) Köster, Felix; Yanchuk, Serhiy; Lüdge, KathyWe show that many delay-based reservoir computers considered in the literature can be characterized by a universal master memory function (MMF). Once computed for two independent parameters, this function provides linear memory capacity for any delay-based single-variable reservoir with small inputs. Moreover, we propose an analytical description of the MMF that enables its efficient and fast computation. Our approach can be applied not only to single-variable delay-based reservoirs governed by known dynamical rules, such as the Mackey–Glass or Stuart–Landau-like systems, but also to reservoirs whose dynamical model is not available.
- 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.
- ItemMultistable jittering in oscillators with pulsatile delayed feedback(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Klinshov, Vladimir; Lücken, Leonhard; Shchapin, Dmitry; Nekorkin, Vladimir; Yanchuk, SerhiyOscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called jittering regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the multi-jitter bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.
- ItemNoise 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, SerhiySpike 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.
- ItemNoise-induced switching in two adaptively coupled excitable systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bačić, Iva; Yanchuk, Serhiy; Wolfrum, Matthias; Franović, IgorWe demonstrate that the interplay of noise and plasticity gives rise to slow stochastic fluctuations in a system of two adaptively coupled active rotators with excitable local dynamics. Depending on the adaptation rate, two qualitatively different types of switching behavior are observed. For slower adaptation, one finds alternation between two modes of noise-induced oscillations, whereby the modes are distinguished by the different order of spiking between the units. In case of faster adaptation, the system switches between the metastable states derived from coexisting attractors of the corresponding deterministic system, whereby the phases exhibit a bursting-like behavior. The qualitative features of the switching dynamics are analyzed within the framework of fast-slow analysis.
- ItemOn the stability of periodic orbits in delay equations with large delay(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Sieber, Jan; Wolfrum, Matthias; Lichtner, Mark; Yanchuk, SerhiyWe prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay
- ItemOscillatory instability in systems with delay(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Wolfrum, Matthias; Yanchuk, SerhiyAny 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.