TY - JOUR A1 - Laing, Carlo R. A1 - Omel'chenko, Oleh T1 - Moving bumps in theta neuron networks JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We consider large networks of theta neurons on a ring, synaptically coupled with an asymmetric kernel. Such networks support stable "bumps" of activity, which move along the ring if the coupling kernel is asymmetric. We investigate the effects of the kernel asymmetry on the existence, stability, and speed of these moving bumps using continuum equations formally describing infinite networks. Depending on the level of heterogeneity within the network, we find complex sequences of bifurcations as the amount of asymmetry is varied, in strong contrast to the behavior of a classical neural field model. Y1 - 2020 U6 - https://doi.org/10.1063/1.5143261 SN - 1054-1500 SN - 1089-7682 VL - 30 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Omel'chenko, Oleh A1 - Ocampo-Espindola, Jorge Luis A1 - Kiss, István Z. T1 - Asymmetry-induced isolated fully synchronized state in coupled oscillator populations JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - A symmetry-breaking mechanism is investigated that creates bistability between fully and partially synchronized states in oscillator networks. Two populations of oscillators with unimodal frequency distribution and different amplitudes, in the presence of weak global coupling, are shown to simplify to a modular network with asymmetrical coupling. With increasing the coupling strength, a synchronization transition is observed with an isolated fully synchronized state. The results are interpreted theoretically in the thermodynamic limit and confirmed in experiments with chemical oscillators. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.L022202 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 2 PB - American Physical Society CY - Melville, NY ER - TY - THES A1 - Omelchenko, Oleh T1 - Synchronität-und-Unordnung-Muster in Netzwerken gekoppelter Oszillatoren T1 - Patterns of synchrony and disorder in networks of coupled oscillators N2 - Synchronization of coupled oscillators manifests itself in many natural and man-made systems, including cyrcadian clocks, central pattern generators, laser arrays, power grids, chemical and electrochemical oscillators, only to name a few. The mathematical description of this phenomenon is often based on the paradigmatic Kuramoto model, which represents each oscillator by one scalar variable, its phase. When coupled, phase oscillators constitute a high-dimensional dynamical system, which exhibits complex behaviour, ranging from synchronized uniform oscillation to quasiperiodicity and chaos. The corresponding collective rhythms can be useful or harmful to the normal operation of various systems, therefore they have been the subject of much research. Initially, synchronization phenomena have been studied in systems with all-to-all (global) and nearest-neighbour (local) coupling, or on random networks. However, in recent decades there has been a lot of interest in more complicated coupling structures, which take into account the spatially distributed nature of real-world oscillator systems and the distance-dependent nature of the interaction between their components. Examples of such systems are abound in biology and neuroscience. They include spatially distributed cell populations, cilia carpets and neural networks relevant to working memory. In many cases, these systems support a rich variety of patterns of synchrony and disorder with remarkable properties that have not been observed in other continuous media. Such patterns are usually referred to as the coherence-incoherence patterns, but in symmetrically coupled oscillator systems they are also known by the name chimera states. The main goal of this work is to give an overview of different types of collective behaviour in large networks of spatially distributed phase oscillators and to develop mathematical methods for their analysis. We focus on the Kuramoto models for one-, two- and three-dimensional oscillator arrays with nonlocal coupling, where the coupling extends over a range wider than nearest neighbour coupling and depends on separation. We use the fact that, for a special (but still quite general) phase interaction function, the long-term coarse-grained dynamics of the above systems can be described by a certain integro-differential equation that follows from the mathematical approach called the Ott-Antonsen theory. We show that this equation adequately represents all relevant patterns of synchrony and disorder, including stationary, periodically breathing and moving coherence-incoherence patterns. Moreover, we show that this equation can be used to completely solve the existence and stability problem for each of these patterns and to reliably predict their main properties in many application relevant situations. N2 - Die Synchronisation von gekoppelten Oszillatoren tritt in vielen natürlichen und künstlichen Systemen auf, beispielsweise bei zirkadianen Uhren, zentralen Mustergeneratoren, Laserarrays, Stromnetzen oder chemischen und elektrochemischen Oszillatoren, um nur einige zu nennen. Die mathematische Beschreibung dieses Phänomens basiert häufig auf dem paradigmatischen Kuramoto-Modell, das jeden Oszillator durch eine skalare Variable, seine Phase, darstellt. Wenn Phasenoszillatoren gekoppelt sind, bilden sie ein hochdimensionales dynamisches System, das ein komplexes Verhalten aufweist, welches von synchronisierter kollektiver Oszillation bis zu Quasiperiodizität und Chaos reicht. Die entsprechenden kollektiven Rhythmen können für den normalen Betrieb verschiedener Systeme nützlich oder schädlich sein, weshalb sie Gegenstand zahlreicher Untersuchungen waren. Anfänglich wurden Synchronisationsphänomene in Systemen mit globaler Mittelfeldkopplung und lokaler Nächster-Nachbar Kopplung oder in komplexen Netzwerken untersucht. In den letzten Jahrzehnten gab es jedoch großes Interesse an anderen Kopplungsstrukturen, die die räumlich verteilte Natur realer Oszillatorsysteme und die entfernungsabhängige Natur der Wechselwirkung zwischen ihren Komponenten berücksichtigen. Sowohl in Bereichen der Biologie als auch der Neurowissenschaften gibt es eine Vielzahl von Beipsieln für solche Systeme. Dazu gehören räumlich verteilte Zellpopulationen, Zilien-Teppiche und neuronale Netze, die für das Arbeitsgedächtnis relevant sind. In vielen Fällen unterstützen diese Systeme eine Vielzahl von Synchronität-und-Unordnung-Mustern mit bemerkenswerten Eigenschaften, die in anderen kontinuierlichen Medien nicht beobachtet wurden. Solche Muster werden üblicherweise als Kohärenz-Inkohärenz-Muster bezeichnet, aber in symmetrisch gekoppelten Oszillatorsystemen sind diese auch unter dem Namen Chimära-Zustände bekannt. Das Hauptziel dieser Arbeit ist es, einen Überblick über verschiedene Arten von kollektivem Verhalten in großen Netzwerken räumlich verteilter Phasenoszillatoren zu geben und mathematische Methoden für deren Analyse zu entwickeln. Wir konzentrieren uns dabei auf die Kuramoto-Modelle für ein-, zwei- und dreidimensionale Oszillator-Arrays mit nichtlokaler Kopplung, wobei sich die Kopplung über einen Bereich erstreckt, welcher breiter ist als die Kopplung zum nächsten Nachbarn und von der Trennung abhängt. Wir verwenden die Tatsache, dass für eine spezielle (aber immer noch recht allgemeine) Phasenwechselwirkungsfunktion die langfristige grobkörnige Dynamik der obigen Systeme durch eine bestimmte Integro-Differentialgleichung beschrieben werden kann. Diese ergibt sich aus dem mathematischen Ansatz namens Ott-Antonsen-Theorie. Wir zeigen, dass diese Gleichung alle relevanten Synchronität-und-Unordnung-Muster angemessen darstellt, einschließlich stationärer, periodisch oszillierender und sich bewegender Kohärenz-Inkohärenz-Muster. Darüber hinaus zeigen wir, dass diese Gleichung verwendet werden kann, um das Existenz- und Stabilitätsproblem für jedes dieser Muster vollständig zu lösen und ihre Haupteigenschaften in vielen anwendungsrelevanten Situationen zuverlässig vorherzusagen. KW - phase oscillators KW - networks KW - synchronization KW - dynamical patterns KW - chimera states KW - Phasenoszillatoren KW - Netzwerke KW - Synchronisation KW - dynamische Muster KW - Chimäre-Zustände Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-535961 ER - TY - JOUR A1 - Franović, Igor A1 - Omel'chenko, Oleh A1 - Wolfrum, Matthias T1 - Bumps, chimera states, and Turing patterns in systems of coupled active rotators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Self-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 nonlocal 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. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.L052201 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 5 PB - American Physical Society CY - College Park ER - TY - GEN A1 - Butuzov, Valentin F. A1 - Nefedov, N. N. A1 - Recke, Lutz A1 - Omel'chenko, Oleh T1 - Partly dissipative system with multizonal initial and boundary layers T2 - Journal of Physics: Conference Series N2 - For a singularly perturbed parabolic - ODE system we construct the asymptotic expansion in the small parameter in the case, when the degenerate equation has a double root. Such systems, which are called partly dissipative reaction-diffusion systems, are used to model various natural processes, including the signal transmission along axons, solid combustion and the kinetics of some chemical reactions. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. The multizonal initial and boundary layers behaviour was stated. Y1 - 2019 U6 - https://doi.org/10.1088/1742-6596/1205/1/012009 SN - 1742-6588 SN - 1742-6596 VL - 1205 PB - IOP Publ. CY - Bristol ER - TY - GEN A1 - Omel'chenko, Oleh T1 - Travelling chimera states in systems of phase oscillators with asymmetric nonlocal coupling T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We study travelling chimera states in a ring of nonlocally coupled heterogeneous (with Lorentzian distribution of natural frequencies) phase oscillators. These states are coherence-incoherence patterns moving in the lateral direction because of the broken reflection symmetry of the coupling topology. To explain the results of direct numerical simulations we consider the continuum limit of the system. In this case travelling chimera states correspond to smooth travelling wave solutions of some integro-differential equation, called the Ott–Antonsen equation, which describes the long time coarse-grained dynamics of the oscillators. Using the Lyapunov–Schmidt reduction technique we suggest a numerical approach for the continuation of these travelling waves. Moreover, we perform their linear stability analysis and show that travelling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated travelling chimera states. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1169 KW - chimera states KW - nonlocally coupled phase oscillators KW - Ott–Antonsen equation KW - forced symmetry breaking KW - travelling waves KW - continuation KW - stability Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-518141 SN - 1866-8372 IS - 2 SP - 611 EP - 642 ER - TY - JOUR A1 - Omel'chenko, Oleh T1 - Mathematical framework for breathing chimera states JF - Journal of nonlinear science N2 - About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique. KW - Coupled oscillators KW - Breathing chimera states KW - Coherence-incoherence KW - patterns KW - Ott-Antonsen equation KW - Periodic solutions KW - Stability Y1 - 2022 U6 - https://doi.org/10.1007/s00332-021-09779-1 SN - 0938-8974 SN - 1432-1467 VL - 32 IS - 2 PB - Springer CY - New York ER - TY - JOUR A1 - Ocampo-Espindola, Jorge Luis A1 - Omel'chenko, Oleh A1 - Kiss, Istvan Z. T1 - Non-monotonic transients to synchrony in Kuramoto networks and electrochemical oscillators JF - Journal of physics. Complexity N2 - We performed numerical simulations with the Kuramoto model and experiments with oscillatory nickel electrodissolution to explore the dynamical features of the transients from random initial conditions to a fully synchronized (one-cluster) state. The numerical simulations revealed that certain networks (e.g., globally coupled or dense Erdos-Renyi random networks) showed relatively simple behavior with monotonic increase of the Kuramoto order parameter from the random initial condition to the fully synchronized state and that the transient times exhibited a unimodal distribution. However, some modular networks with bridge elements were identified which exhibited non-monotonic variation of the order parameter with local maximum and/or minimum. In these networks, the histogram of the transients times became bimodal and the mean transient time scaled well with inverse of the magnitude of the second largest eigenvalue of the network Laplacian matrix. The non-monotonic transients increase the relative standard deviations from about 0.3 to 0.5, i.e., the transient times became more diverse. The non-monotonic transients are related to generation of phase patterns where the modules are synchronized but approximately anti-phase to each other. The predictions of the numerical simulations were demonstrated in a population of coupled oscillatory electrochemical reactions in global, modular, and irregular tree networks. The findings clarify the role of network structure in generation of complex transients that can, for example, play a role in intermittent desynchronization of the circadian clock due to external cues or in deep brain stimulations where long transients are required after a desynchronization stimulus. KW - synchronization KW - networks KW - Kuramoto model KW - electrochemistry KW - chemical KW - oscillations Y1 - 2021 U6 - https://doi.org/10.1088/2632-072X/abe109 SN - 2632-072X VL - 2 IS - 1 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Bataille-Gonzalez, Martin A1 - Clerc, Marcel G. A1 - Omel'chenko, Oleh T1 - Moving spiral wave chimeras JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We consider a two-dimensional array of heterogeneous nonlocally coupled phase oscillators on a flat torus and study the bound states of two counter-rotating spiral chimeras, shortly two-core spiral chimeras, observed in this system. In contrast to other known spiral chimeras with motionless incoherent cores, the two-core spiral chimeras typically show a drift motion. Due to this drift, their incoherent cores become spatially modulated and develop specific fingerprint patterns of varying synchrony levels. In the continuum limit of infinitely many oscillators, the two-core spiral chimeras can be studied using the Ott-Antonsen equation. Numerical analysis of this equation allows us to reveal the stability region of different spiral chimeras, which we group into three main classes-symmetric, asymmetric, and meandering spiral chimeras. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.L022203 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Omel'chenko, Oleh A1 - Tél, Tamás T1 - Focusing on transient chaos JF - Journal of Physics: Complexity N2 - Recent advances in the field of complex, transiently chaotic dynamics are reviewed, based on the results published in the focus issue of J. Phys. Complex. on this topic. One group of achievements concerns network dynamics where transient features are intimately related to the degree and stability of synchronization, as well as to the network topology. A plethora of various applications of transient chaos are described, ranging from the collective motion of active particles, through the operation of power grids, cardiac arrhythmias, and magnetohydrodynamical dynamos, to the use of machine learning to predict time evolutions. Nontraditional forms of transient chaos are also explored, such as the temporal change of the chaoticity in the transients (called doubly transient chaos), as well as transients in systems subjected to parameter drift, the paradigm of which is climate change. KW - transient chaos KW - network dynamics KW - applications KW - doubly transient chaos KW - systems subjected to parameter drift Y1 - 2022 U6 - https://doi.org/10.1088/2632-072X/ac5566 SN - 2632-072X VL - 3 IS - 1 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Omel'chenko, Oleh A1 - Laing, Carlo R. T1 - Collective states in a ring network of theta neurons JF - Proceedings of the Royal Society of London. Series A, Mathematical, physical and engineering sciences N2 - We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results. KW - theta neurons KW - neural networks KW - bumps Y1 - 2022 U6 - https://doi.org/10.1098/rspa.2021.0817 SN - 1364-5021 SN - 1471-2946 VL - 478 IS - 2259 PB - Royal Society CY - London ER -