@phdthesis{Zheng2021,
  author    = {Zheng, Chunming},
  title     = {Bursting and synchronization in noisy oscillatory systems},
  doi       = {10.25932/publishup-50019},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-500199},
  school      = {Universit{\"a}t Potsdam},
  pages     = {iv, 87},
  year      = {2021},
  abstract  = {Noise is ubiquitous in nature and usually results in rich dynamics in stochastic systems such as oscillatory systems, which exist in such various fields as physics, biology and complex networks. The correlation and synchronization of two or many oscillators are widely studied topics in recent years. In this thesis, we mainly investigate two problems, i.e., the stochastic bursting phenomenon in noisy excitable systems and synchronization in a three-dimensional Kuramoto model with noise. Stochastic bursting here refers to a sequence of coherent spike train, where each spike has random number of followers due to the combined effects of both time delay and noise. Synchronization, as a universal phenomenon in nonlinear dynamical systems, is well illustrated in the Kuramoto model, a prominent model in the description of collective motion. In the first part of this thesis, an idealized point process, valid if the characteristic timescales in the problem are well separated, is used to describe statistical properties such as the power spectral density and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate, and the probability to induce a spike during the delay action can be calculated from the solutions of a stationary and a forced Fokker-Planck equation. We extend it to the delay-coupled case and derive analytically the statistics of the spikes in each neuron, the pairwise correlations between any two neurons, and the spectrum of the total output from the network. In the second part, we investigate the three-dimensional noisy Kuramoto model, which can be used to describe the synchronization in a swarming model with helical trajectory. In the case without natural frequency, the Kuramoto model can be connected with the Vicsek model, which is widely studied in collective motion and swarming of active matter. We analyze the linear stability of the incoherent state and derive the critical coupling strength above which the incoherent state loses stability. In the limit of no natural frequency, an exact self-consistent equation of the mean field is derived and extended straightforward to any high-dimensional case.},
  language  = {en}
}
@phdthesis{Yeldesbay2014,
  author    = {Yeldesbay, Azamat},
  title     = {Complex regimes of synchronization},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-73348},
  school      = {Universit{\"a}t Potsdam},
  pages     = {ii, 60},
  year      = {2014},
  abstract  = {Synchronization is a fundamental phenomenon in nature. It can be considered as a general property of self-sustained oscillators to adjust their rhythm in the presence of an interaction. In this work we investigate complex regimes of synchronization phenomena by means of theoretical analysis, numerical modeling, as well as practical analysis of experimental data. As a subject of our investigation we consider chimera state, where due to spontaneous symmetry-breaking of an initially homogeneous oscillators lattice split the system into two parts with different dynamics. Chimera state as a new synchronization phenomenon was first found in non-locally coupled oscillators system, and has attracted a lot of attention in the last decade. However, the recent studies indicate that this state is also possible in globally coupled systems. In the first part of this work, we show under which conditions the chimera-like state appears in a system of globally coupled identical oscillators with intrinsic delayed feedback. The results of the research explain how initially monostable oscillators became effectivly bistable in the presence of the coupling and create a mean field that sustain the coexistence of synchronized and desynchronized states. Also we discuss other examples, where chimera-like state appears due to frequency dependence of the phase shift in the bistable system. In the second part, we make further investigation of this topic by modeling influence of an external periodic force to an oscillator with intrinsic delayed feedback. We made stability analysis of the synchronized state and constructed Arnold tongues. The results explain formation of the chimera-like state and hysteric behavior of the synchronization area. Also, we consider two sets of parameters of the oscillator with symmetric and asymmetric Arnold tongues, that correspond to mono- and bi-stable regimes of the oscillator. In the third part, we demonstrate the results of the work, which was done in collaboration with our colleagues from Psychology Department of University of Potsdam. The project aimed to study the effect of the cardiac rhythm on human perception of time using synchronization analysis. From our part, we made a statistical analysis of the data obtained from the conducted experiment on free time interval reproduction task. We examined how ones heartbeat influences the time perception and searched for possible phase synchronization between heartbeat cycles and time reproduction responses. The findings support the prediction that cardiac cycles can serve as input signals, and is used for reproduction of time intervals in the range of several seconds.},
  language  = {en}
}
@phdthesis{Vlasov2015,
  author    = {Vlasov, Vladimir},
  title     = {Synchronization of oscillatory networks in terms of global variables},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78182},
  school      = {Universit{\"a}t Potsdam},
  pages     = {82},
  year      = {2015},
  abstract  = {Synchronization of large ensembles of oscillators is an omnipresent phenomenon observed in different fields of science like physics, engineering, life sciences, etc. The most simple setup is that of globally coupled phase oscillators, where all the oscillators contribute to a global field which acts on all oscillators. This formulation of the problem was pioneered by Winfree and Kuramoto. Such a setup gives a possibility for the analysis of these systems in terms of global variables. In this work we describe nontrivial collective dynamics in oscillator populations coupled via mean fields in terms of global variables. We consider problems which cannot be directly reduced to standard Kuramoto and Winfree models. In the first part of the thesis we adopt a method introduced by Watanabe and Strogatz. The main idea is that the system of identical oscillators of particular type can be described by a low-dimensional system of global equations. This approach enables us to perform a complete analytical analysis for a special but vast set of initial conditions. Furthermore, we show how the approach can be expanded for some nonidentical systems. We apply the Watanabe-Strogatz approach to arrays of Josephson junctions and systems of identical phase oscillators with leader-type coupling. In the next parts of the thesis we consider the self-consistent mean-field theory method that can be applied to general nonidentical globally coupled systems of oscillators both with or without noise. For considered systems a regime, where the global field rotates uniformly, is the most important one. With the help of this approach such solutions of the self-consistency equation for an arbitrary distribution of frequencies and coupling parameters can be found analytically in the parametric form, both for noise-free and noisy cases. We apply this method to deterministic Kuramoto-type model with generic coupling and an ensemble of spatially distributed oscillators with leader-type coupling. Furthermore, with the proposed self-consistent approach we fully characterize rotating wave solutions of noisy Kuramoto-type model with generic coupling and an ensemble of noisy oscillators with bi-harmonic coupling. Whenever possible, a complete analysis of global dynamics is performed and compared with direct numerical simulations of large populations.},
  language  = {en}
}
@phdthesis{Toenjes2007,
  author    = {T{\"o}njes, Ralf},
  title     = {Pattern formation through synchronization in systems of nonidentical autonomous oscillators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15973},
  school      = {Universit{\"a}t Potsdam},
  year      = {2007},
  abstract  = {This work is concerned with the spatio-temporal structures that emerge when non-identical, diffusively coupled oscillators synchronize. It contains analytical results and their confirmation through extensive computer simulations. We use the Kuramoto model which reduces general oscillatory systems to phase dynamics. The symmetry of the coupling plays an important role for the formation of patterns. We have studied the ordering influence of an asymmetry (non-isochronicity) in the phase coupling function on the phase profile in synchronization and the intricate interplay between this asymmetry and the frequency heterogeneity in the system. The thesis is divided into three main parts. Chapter 2 and 3 introduce the basic model of Kuramoto and conditions for stable synchronization. In Chapter 4 we characterize the phase profiles in synchronization for various special cases and in an exponential approximation of the phase coupling function, which allows for an analytical treatment. Finally, in the third part (Chapter 5) we study the influence of non-isochronicity on the synchronization frequency in continuous, reaction diffusion systems and discrete networks of oscillators.},
  language  = {en}
}
@phdthesis{RomanoBlasco2004,
  author    = {Romano Blasco, M. Carmen},
  title     = {Synchronization analysis by means of recurrences in phase space},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001756},
  school      = {Universit{\"a}t Potsdam},
  year      = {2004},
  abstract  = {Die t{\"a}gliche Erfahrung zeigt uns, daß bei vielen physikalischen Systemen kleine {\"A}nderungen in den Anfangsbedingungen auch zu kleinen {\"A}nderungen im Verhalten des Systems f{\"u}hren. Wenn man z.B. das Steuerrad beim Auto fahren nur ein wenig zur Seite dreht, unterscheidet sich die Richtung des Wagens auch nur wenig von der urspr{\"u}nglichen Richtung. Aber es gibt auch Situationen, f{\"u}r die das Gegenteil dieser Regel zutrifft. Die Folge von Kopf und Zahl, die wir erhalten, wenn wir eine M{\"u}nze werfen, zeigt ein irregul{\"a}res oder chaotisches Zeitverhalten, da winzig kleine {\"A}nderungen in den Anfangsbedingungen, die z.B. durch leichte Drehung der Hand hervorgebracht werden, zu vollkommen verschiedenen Resultaten f{\"u}hren. In den letzten Jahren hat man sehr viele nichtlineare Systeme mit schnellen Rechnern untersucht und festgestellt, daß eine sensitive Abh{\"a}ngigkeit von den Anfangsbedingungen, die zu einem chaotischen Verhalten f{\"u}hrt, keinesfalls die Ausnahme darstellt, sondern eine typische Eigenschaft vieler Systeme ist. Obwohl chaotische Systeme kleinen {\"A}nderungen in den Anfangsbedingungen gegen{\"u}ber sehr empfindlich reagieren, k{\"o}nnen sie synchronisieren wenn sie durch eine gemeinsame {\"a}ußere Kraft getrieben werden, oder wenn sie miteinander gekoppelt sind. Das heißt, sie vergessen ihre Anfangsbedingungen und passen ihre Rhythmen aneinander. Diese Eigenschaft chaotischer Systeme hat viele Anwendungen, wie z.B. das Design von Kommunikationsger{\"a}te und die verschl{\"u}sselte {\"U}bertragung von Mitteilungen. Abgesehen davon, findet man Synchronisation in nat{\"u}rlichen Systemen, wie z.B. das Herz-Atmungssystem, raumverteilte {\"o}kologische Systeme, die Magnetoenzephalographische Aktivit{\"a}t von Parkinson Patienten, etc. In solchen komplexen Systemen ist es nicht trivial Synchronisation zu detektieren und zu quantifizieren. Daher ist es notwendig, besondere mathematische Methoden zu entwickeln, die diese Aufgabe erledigen. Das ist das Ziel dieser Arbeit. Basierend auf dergrundlegenden Idee von Rekurrenzen (Wiederkehr) von Trajektorien dynamischer Systeme, sind verschiedene Maße entwickelt worden, die Synchronisation in chaotischen und komplexen Systemen detektieren. Das Wiederkehr von Trajektorien erlaubt uns Vorhersagen {\"u}ber den zuk{\"u}nftigen Zustand eines Systems zu treffen. Wenn man diese Eigenschaft der Wiederkehr von zwei interagierenden Systemen vergleicht, kann man Schl{\"u}sse {\"u}ber ihre dynamische Anpassung oder Synchronisation ziehen. Ein wichtiger Vorteil der Rekurrenzmaße f{\"u}r Synchronisation ist die Robustheit gegen Rauschen und Instationari{\"a}t. Das erlaubt eine Synchronisationsanalyse in Systemen durchzuf{\"u}hren, die bisher nicht darauf untersucht werden konnten.},
  language  = {en}
}
@phdthesis{Peter2019,
  author    = {Peter, Franziska},
  title     = {Transition to synchrony in finite Kuramoto ensembles},
  doi       = {10.25932/publishup-42916},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-429168},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vi, 93},
  year      = {2019},
  abstract  = {Synchronisation - die Ann{\"a}herung der Rhythmen gekoppelter selbst oszillierender Systeme - ist ein faszinierendes dynamisches Ph{\"a}nomen, das in vielen biologischen, sozialen und technischen Systemen auftritt. Die vorliegende Arbeit befasst sich mit Synchronisation in endlichen Ensembles schwach gekoppelter selbst-erhaltender Oszillatoren mit unterschiedlichen nat{\"u}rlichen Frequenzen. Das Standardmodell f{\"u}r dieses kollektive Ph{\"a}nomen ist das Kuramoto-Modell - unter anderem aufgrund seiner L{\"o}sbarkeit im thermodynamischen Limes unendlich vieler Oszillatoren. {\"A}hnlich einem thermodynamischen Phasen{\"u}bergang zeigt im Fall unendlich vieler Oszillatoren ein Ordnungsparameter den {\"U}bergang von Inkoh{\"a}renz zu einem partiell synchronen Zustand an, in dem ein Teil der Oszillatoren mit einer gemeinsamen Frequenz rotiert. Im endlichen Fall treten Fluktuationen auf. In dieser Arbeit betrachten wir den bisher wenig beachteten Fall von bis zu wenigen hundert Oszillatoren, unter denen vergleichbar starke Fluktuationen auftreten, bei denen aber ein Vergleich zu Frequenzverteilungen im unendlichen Fall m{\"o}glich ist. Zun{\"a}chst definieren wir einen alternativen Ordnungsparameter zur Feststellung einer kollektiven Mode im endlichen Kuramoto-Modell. Dann pr{\"u}fen wir die Abh{\"a}ngigkeit des Synchronisationsgrades und der mittleren Rotationsfrequenz der kollektiven Mode von Eigenschaften der nat{\"u}rlichen Frequenzverteilung f{\"u}r verschiedene Kopplungsst{\"a}rken. Wir stellen dabei zun{\"a}chst numerisch fest, dass der Synchronisationsgrad stark von der Form der Verteilung (gemessen durch die Kurtosis) und die Rotationsfrequenz der kollektiven Mode stark von der Asymmetrie der Verteilung (gemessen durch die Schiefe) der nat{\"u}rlichen Frequenzen abh{\"a}ngt. Beides k{\"o}nnen wir im thermodynamischen Limes analytisch verifizieren. Mit diesen Ergebnissen k{\"o}nnen wir Erkenntnisse anderer Autoren besser verstehen und verallgemeinern. Etwas abseits des roten Fadens dieser Arbeit finden wir außerdem einen analytischen Ausdruck f{\"u}r die Volumenkontraktion im Phasenraum. Der zweite Teil der Arbeit konzentriert sich auf den ordnenden Effekt von Fluktuationen, die durch die Endlichkeit des Systems zustande kommen. Im unendlichen Modell sind die Oszillatoren eindeutig in koh{\"a}rent und inkoh{\"a}rent und damit in geordnet und ungeordnet getrennt. Im endlichen Fall k{\"o}nnen die auftretenden Fluktuationen zus{\"a}tzliche Ordnung unter den asynchronen Oszillatoren erzeugen. Das grundlegende Prinzip, die rauschinduzierte Synchronisation, ist aus einer Reihe von Publikationen bekannt. Unter den gekoppelten Oszillatoren n{\"a}hern sich die Phasen aufgrund der Fluktuationen des Ordnungsparameters an, wie wir einerseits direkt numerisch zeigen und andererseits mit einem Synchronisationsmaß aus der gerichteten Statistik zwischen Paaren passiver Oszillatoren nachweisen. Wir bestimmen die Abh{\"a}ngigkeit dieses Synchronisationsmaßes vom Verh{\"a}ltnis von paarweiser nat{\"u}rlicher Frequenzdifferenz zur Varianz der Fluktuationen. Dabei finden wir eine gute {\"U}bereinstimmung mit einem einfachen analytischen Modell, in welchem wir die deterministischen Fluktuationen des Ordnungsparameters durch weißes Rauschen ersetzen.},
  language  = {en}
}
@phdthesis{Omelchenko2021,
  author    = {Omelchenko, Oleh},
  title     = {Synchronit{\"a}t-und-Unordnung-Muster in Netzwerken gekoppelter Oszillatoren},
  doi       = {10.25932/publishup-53596},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-535961},
  school      = {Universit{\"a}t Potsdam},
  pages     = {152},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}
@phdthesis{MontbrioiFairen2004,
  author    = {Montbri{\´o} i Fairen, Ernest},
  title     = {Synchronization in ensembles of nonisochronous oscillators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001492},
  school      = {Universit{\"a}t Potsdam},
  year      = {2004},
  abstract  = {Diese Arbeit analysiert Synchronisationsphaenomene, die in grossen Ensembles von interagierenden Oszillatoren auftauchen. Im speziellen werden die Effekte von Nicht-Isochronizitaet (die Abhaengigkeit der Frequenz von der Amplitude des Oszillators) auf den makroskopischen Uebergang zur Synchronisation im Detail studiert. Die neu gefundenen Phaenomene (Anomale Synchronisation) werden sowohl in Populationen von Oszillatoren als auch zwischen Oszillator-Ensembles untersucht.},
  language  = {en}
}
@phdthesis{Kucklaender2006,
  author    = {Kuckl{\"a}nder, Nina},
  title     = {Synchronization via correlated noise and automatic control in ecological systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-10826},
  school      = {Universit{\"a}t Potsdam},
  year      = {2006},
  abstract  = {<img src="http://vg00.met.vgwort.de/na/806c85cec18906a64e06" width="1" height="1" alt=""> Subject of this work is the possibility to synchronize nonlinear systems via correlated noise and automatic control. The thesis is divided into two parts. The first part is motivated by field studies on feral sheep populations on two islands of the St. Kilda archipelago, which revealed strong correlations due to environmental noise. For a linear system the population correlation equals the noise correlation (Moran effect). But there exists no systematic examination of the properties of nonlinear maps under the influence of correlated noise. Therefore, in the first part of this thesis the noise-induced correlation of logistic maps is systematically examined. For small noise intensities it can be shown analytically that the correlation of quadratic maps in the fixed-point regime is always smaller than or equal to the noise correlation. In the period-2 regime a Markov model explains qualitatively the main dynamical characteristics. Furthermore, two different mechanisms are introduced which lead to a higher correlation of the systems than the environmental correlation. The new effect of "correlation resonance" is described, i. e. the correlation yields a maximum depending on the noise intensity. In the second part of the thesis an automatic control method is presented which synchronizes different systems in a robust way. This method is inspired by phase-locked loops and is based on a feedback loop with a differential control scheme, which allows to change the phases of the controlled systems. The effectiveness of the approach is demonstrated for controlled phase synchronization of regular oscillators and foodweb models.},
  subject      = {Markov-Prozess},
  language  = {en}
}
@phdthesis{Goldobin2007,
  author    = {Goldobin, Denis S.},
  title     = {Coherence and synchronization of noisy-driven oscillators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15047},
  school      = {Universit{\"a}t Potsdam},
  year      = {2007},
  abstract  = {In the present dissertation paper we study problems related to synchronization phenomena in the presence of noise which unavoidably appears in real systems. One part of the work is aimed at investigation of utilizing delayed feedback to control properties of diverse chaotic dynamic and stochastic systems, with emphasis on the ones determining predisposition to synchronization. Other part deals with a constructive role of noise, i.e. its ability to synchronize identical self-sustained oscillators. First, we demonstrate that the coherence of a noisy or chaotic self-sustained oscillator can be efficiently controlled by the delayed feedback. We develop the analytical theory of this effect, considering noisy systems in the Gaussian approximation. Possible applications of the effect for the synchronization control are also discussed. Second, we consider synchrony of limit cycle systems (in other words, self-sustained oscillators) driven by identical noise. For weak noise and smooth systems we proof the purely synchronizing effect of noise. For slightly different oscillators and/or slightly nonidentical driving, synchrony becomes imperfect, and this subject is also studied. Then, with numerics we show moderate noise to be able to lead to desynchronization of some systems under certain circumstances. For neurons the last effect means "antireliability" (the "reliability" property of neurons is treated to be important from the viewpoint of information transmission functions), and we extend our investigation to neural oscillators which are not always limit cycle ones. Third, we develop a weakly nonlinear theory of the Kuramoto transition (a transition to collective synchrony) in an ensemble of globally coupled oscillators in presence of additional time-delayed coupling terms. We show that a linear delayed feedback not only controls the transition point, but effectively changes the nonlinear terms near the transition. A purely nonlinear delayed coupling does not affect the transition point, but can reduce or enhance the amplitude of collective oscillations.},
  language  = {en}
}