@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{Teichmann2021, author = {Teichmann, Erik}, title = {Partial synchronization in coupled systems with repulsive and attractive interaction}, doi = {10.25932/publishup-52894}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-528943}, school = {Universit{\"a}t Potsdam}, pages = {x, 96}, year = {2021}, abstract = {Partial synchronous states exist in systems of coupled oscillators between full synchrony and asynchrony. They are an important research topic because of their variety of different dynamical states. Frequently, they are studied using phase dynamics. This is a caveat, as phase dynamics are generally obtained in the weak coupling limit of a first-order approximation in the coupling strength. The generalization to higher orders in the coupling strength is an open problem. Of particular interest in the research of partial synchrony are systems containing both attractive and repulsive coupling between the units. Such a mix of coupling yields very specific dynamical states that may help understand the transition between full synchrony and asynchrony. This thesis investigates partial synchronous states in mixed-coupling systems. First, a method for higher-order phase reduction is introduced to observe interactions beyond the pairwise one in the first-order phase description, hoping that these may apply to mixed-coupling systems. This new method for coupled systems with known phase dynamics of the units gives correct results but, like most comparable methods, is computationally expensive. It is applied to three Stuart-Landau oscillators coupled in a line with a uniform coupling strength. A numerical method is derived to verify the analytical results. These results are interesting but give importance to simpler phase models that still exhibit exotic states. Such simple models that are rarely considered are Kuramoto oscillators with attractive and repulsive interactions. Depending on how the units are coupled and the frequency difference between the units, it is possible to achieve many different states. Rich synchronization dynamics, such as a Bellerophon state, are observed when considering a Kuramoto model with attractive interaction in two subpopulations (groups) and repulsive interactions between groups. In two groups, one attractive and one repulsive, of identical oscillators with a frequency difference, an interesting solitary state appears directly between full and partial synchrony. This system can be described very well analytically.}, language = {en} } @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{Gong2019, author = {Gong, Chen Chris}, title = {Synchronization of coupled phase oscillators}, doi = {10.25932/publishup-48752}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-487522}, school = {Universit{\"a}t Potsdam}, pages = {xvii, 115}, year = {2019}, abstract = {Oscillatory systems under weak coupling can be described by the Kuramoto model of phase oscillators. Kuramoto phase oscillators have diverse applications ranging from phenomena such as communication between neurons and collective influences of political opinions, to engineered systems such as Josephson Junctions and synchronized electric power grids. This thesis includes the author's contribution to the theoretical framework of coupled Kuramoto oscillators and to the understanding of non-trivial N-body dynamical systems via their reduced mean-field dynamics. The main content of this thesis is composed of four parts. First, a partially integrable theory of globally coupled identical Kuramoto oscillators is extended to include pure higher-mode coupling. The extended theory is then applied to a non-trivial higher-mode coupled model, which has been found to exhibit asymmetric clustering. Using the developed theory, we could predict a number of features of the asymmetric clustering with only information of the initial state provided. The second part consists of an iterated discrete-map approach to simulate phase dynamics. The proposed map --- a Moebius map --- not only provides fast computation of phase synchronization, it also precisely reflects the underlying group structure of the dynamics. We then compare the iterated-map dynamics and various analogous continuous-time dynamics. We are able to replicate known phenomena such as the synchronization transition of the Kuramoto-Sakaguchi model of oscillators with distributed natural frequencies, and chimera states for identical oscillators under non-local coupling. The third part entails a particular model of repulsively coupled identical Kuramoto-Sakaguchi oscillators under common random forcing, which can be shown to be partially integrable. Via both numerical simulations and theoretical analysis, we determine that such a model cannot exhibit stationary multi-cluster states, contrary to the numerical findings in previous literature. Through further investigation, we find that the multi-clustering states reported previously occur due to the accumulation of discretization errors inherent in the integration algorithms, which introduce higher-mode couplings into the model. As a result, the partial integrability condition is violated. Lastly, we derive the microscopic cross-correlation of globally coupled non-identical Kuramoto oscillators under common fluctuating forcing. The effect of correlation arises naturally in finite populations, due to the non-trivial fluctuations of the meanfield. In an idealized model, we approximate the finite-sized fluctuation by a Gaussian white noise. The analytical approximation qualitatively matches the measurements in numerical experiments, however, due to other periodic components inherent in the fluctuations of the mean-field there still exist significant inconsistencies.}, 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} } @book{SchreiberKrahnIngallsetal.2016, author = {Schreiber, Robin and Krahn, Robert and Ingalls, Daniel H. H. and Hirschfeld, Robert}, title = {Transmorphic}, number = {110}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-387-9}, issn = {1613-5652}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-98300}, publisher = {Universit{\"a}t Potsdam}, pages = {100}, year = {2016}, abstract = {Defining Graphical User Interfaces (GUIs) through functional abstractions can reduce the complexity that arises from mutable abstractions. Recent examples, such as Facebook's React GUI framework have shown, how modelling the view as a functional projection from the application state to a visual representation can reduce the number of interacting objects and thus help to improve the reliabiliy of the system. This however comes at the price of a more rigid, functional framework where programmers are forced to express visual entities with functional abstractions, detached from the way one intuitively thinks about the physical world. In contrast to that, the GUI Framework Morphic allows interactions in the graphical domain, such as grabbing, dragging or resizing of elements to evolve an application at runtime, providing liveness and directness in the development workflow. Modelling each visual entity through mutable abstractions however makes it difficult to ensure correctness when GUIs start to grow more complex. Furthermore, by evolving morphs at runtime through direct manipulation we diverge more and more from the symbolic description that corresponds to the morph. Given that both of these approaches have their merits and problems, is there a way to combine them in a meaningful way that preserves their respective benefits? As a solution for this problem, we propose to lift Morphic's concept of direct manipulation from the mutation of state to the transformation of source code. In particular, we will explore the design, implementation and integration of a bidirectional mapping between the graphical representation and a functional and declarative symbolic description of a graphical user interface within a self hosted development environment. We will present Transmorphic, a functional take on the Morphic GUI Framework, where the visual and structural properties of morphs are defined in a purely functional, declarative fashion. In Transmorphic, the developer is able to assemble different morphs at runtime through direct manipulation which is automatically translated into changes in the code of the application. In this way, the comprehensiveness and predictability of direct manipulation can be used in the context of a purely functional GUI, while the effects of the manipulation are reflected in a medium that is always in reach for the programmer and can even be used to incorporate the source transformations into the source files of the application.}, 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{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{Schaefer2014, author = {Schaefer, Laura V.}, title = {Synchronisationsph{\"a}nomene myotendin{\"o}ser Oszillationen interagierender neuromuskul{\"a}rer Systeme}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72445}, school = {Universit{\"a}t Potsdam}, year = {2014}, abstract = {Muskeln oszillieren nachgewiesener Weise mit einer Frequenz um 10 Hz. Doch was geschieht mit myofaszialen Oszillationen, wenn zwei neuromuskul{\"a}re Systeme interagieren? Die Dissertation widmet sich dieser Fragestellung bei isometrischer Interaktion. W{\"a}hrend der Testmessungen ergaben sich Hinweise f{\"u}r das Vorhandensein von m{\"o}glicherweise zwei verschiedenen Formen der Isometrie. Arbeiten zwei Personen isometrisch gegeneinander, k{\"o}nnen subjektiv zwei Modi eingenommen werden: man kann entweder isometrisch halten - der Kraft des Partners widerstehen - oder isometrisch dr{\"u}cken - gegen den isometrischen Widerstand des Partners arbeiten. Daher wurde zus{\"a}tzlich zu den Messungen zur Interaktion zweier Personen an einzelnen Individuen gepr{\"u}ft, ob m{\"o}glicherweise zwei Formen der Isometrie existieren. Die Promotion besteht demnach aus zwei inhaltlich und methodisch getrennten Teilen: I „Single-Isometrie" und II „Paar-Isometrie". F{\"u}r Teil I wurden mithilfe eines pneumatisch betriebenen Systems die hypothetischen Messmodi Halten und Dr{\"u}cken w{\"a}hrend isometrischer Aktion untersucht. Bei n = 10 Probanden erfolgte parallel zur Aufzeichnung des Drucksignals w{\"a}hrend der Messungen die Erfassung der Kraft (DMS) und der Beschleunigung sowie die Aufnahme der mechanischen Muskeloszillationen folgender myotendin{\"o}ser Strukturen via Mechanomyo- (MMG) bzw. Mechanotendografie (MTG): M. triceps brachii (MMGtri), Trizepssehne (MTGtri), M. obliquus externus abdominis (MMGobl). Pro Proband wurden bei 80 \% der MVC sowohl sechs 15-Sekunden-Messungen (jeweils drei im haltenden bzw. dr{\"u}ckenden Modus; Pause: 1 Minute) als auch vier Erm{\"u}dungsmessungen (jeweils zwei im haltenden bzw. dr{\"u}ckenden Modus; Pause: 2 Minuten) durchgef{\"u}hrt. Zum Vergleich der Messmodi Halten und Dr{\"u}cken wurden die Amplituden der myofaszialen Oszillationen sowie die Kraftausdauer herangezogen. Signifikante Unterschiede zwischen dem haltenden und dem dr{\"u}ckenden Modus zeigten sich insbesondere im Bereich der Erm{\"u}dungscharakteristik. So lassen Probanden im haltenden Modus signifikant fr{\"u}her nach als im dr{\"u}ckenden Modus (t(9) = 3,716; p = .005). Im dr{\"u}ckenden Modus macht das l{\"a}ngste isometrische Plateau durchschnittlich 59,4 \% der Gesamtdauer aus, im haltenden sind es 31,6 \% (t(19) = 5,265, p = .000). Die Amplituden der Single-Isometrie-Messungen unterscheiden sich nicht signifikant. Allerdings variieren die Amplituden des MMGobl zwischen den Messungen im dr{\"u}ckenden Modus signifikant st{\"a}rker als im haltenden Modus. Aufgrund dieser teils signifikanten Unterschiede zwischen den beiden Messmodi wurde dieses Setting auch im zweiten Teil „Paar-Isometrie" ber{\"u}cksichtigt. Dort wurden n = 20 Probanden - eingeteilt in zehn gleichgeschlechtliche Paare - w{\"a}hrend isometrischer Interaktion untersucht. Die Sensorplatzierung erfolgte analog zu Teil I. Die Oszillationen der erfassten MTG- sowie MMG-Signale wurden u.a. mit Algorithmen der Nichtlinearen Dynamik auf ihre Koh{\"a}renz hin untersucht. Durch die Paar-Isometrie-Messungen zeigte sich, dass die Muskeln und die Sehnen beider neuromuskul{\"a}rer Systeme bei Interaktion im bekannten Frequenzbereich von 10 Hz oszillieren. Außerdem waren sie in der Lage, sich bei Interaktion so aufeinander abzustimmen, dass sich eine signifikante Koh{\"a}renz entwickelte, die sich von Zufallspaarungen signifikant unterscheidet (Patchanzahl: t(29) = 3,477; p = .002; Summe der 4 l{\"a}ngsten Patches: t(29) = 7,505; p = .000). Es wird der Schluss gezogen, dass neuromuskul{\"a}re Komplement{\"a}rpartner in der Lage sind, sich im Sinne koh{\"a}renten Verhaltens zu synchronisieren. Bez{\"u}glich der Parameter zur Untersuchung der m{\"o}glicherweise vorhandenen zwei Formen der Isometrie zeigte sich bei den Paar-Isometrie-Messungen zwischen Halten und Dr{\"u}cken ein signifikanter Unterschied bei der Erm{\"u}dungscharakteristik sowie bez{\"u}glich der Amplitude der MMGobl. Die Ergebnisse beider Teilstudien best{\"a}rken die Hypothese, dass zwei Formen der Isometrie existieren. Fraglich ist, ob man {\"u}berhaupt von Isometrie sprechen kann, da jede isometrische Muskelaktion aus feinen Oszillationen besteht, die eine per Definition postulierte Isometrie ausschließen. Es wird der Vorschlag unterbreitet, die Isometrie durch den Begriff der Hom{\"o}ometrie auszutauschen. Die Ergebnisse der Paar-Isometrie-Messungen zeigen u.a., dass neuromuskul{\"a}re Systeme in der Lage sind, ihre myotendin{\"o}sen Oszillationen so aufeinander abzustimmen, dass koh{\"a}rentes Verhalten entsteht. Es wird angenommen, dass hierzu beide neuromuskul{\"a}ren Systeme funktionell intakt sein m{\"u}ssen. Das Verfahren k{\"o}nnte f{\"u}r die Diagnostik funktioneller St{\"o}rungen relevant werden.}, language = {de} } @phdthesis{Fischer2014, author = {Fischer, Jost Leonhardt}, title = {Nichtlineare Kopplungsmechanismen akustischer Oszillatoren am Beispiel der Synchronisation von Orgelpfeifen}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71975}, school = {Universit{\"a}t Potsdam}, year = {2014}, abstract = {In dieser Arbeit werden nichtlineare Kopplungsmechanismen von akustischen Oszillatoren untersucht, die zu Synchronisation f{\"u}hren k{\"o}nnen. Aufbauend auf die Fragestellungen vorangegangener Arbeiten werden mit Hilfe theoretischer und experimenteller Studien sowie mit Hilfe numerischer Simulationen die Elemente der Tonentstehung in der Orgelpfeife und die Mechanismen der gegenseitigen Wechselwirkung von Orgelpfeifen identifiziert. Daraus wird erstmalig ein vollst{\"a}ndig auf den aeroakustischen und fluiddynamischen Grundprinzipien basierendes nichtlinear gekoppeltes Modell selbst-erregter Oszillatoren f{\"u}r die Beschreibung des Verhaltens zweier wechselwirkender Orgelpfeifen entwickelt. Die durchgef{\"u}hrten Modellrechnungen werden mit den experimentellen Befunden verglichen. Es zeigt sich, dass die Tonentstehung und die Kopplungsmechanismen von Orgelpfeifen durch das entwickelte Oszillatormodell in weiten Teilen richtig beschrieben werden. Insbesondere kann damit die Ursache f{\"u}r den nichtlinearen Zusammenhang von Kopplungsst{\"a}rke und Synchronisation des gekoppelten Zwei-Pfeifen Systems, welcher sich in einem nichtlinearen Verlauf der Arnoldzunge darstellt, gekl{\"a}rt werden. Mit den gewonnenen Erkenntnissen wird der Einfluss des Raumes auf die Tonentstehung bei Orgelpfeifen betrachtet. Daf{\"u}r werden numerische Simulationen der Wechselwirkung einer Orgelpfeife mit verschiedenen Raumgeometrien, wie z. B. ebene, konvexe, konkave, und gezahnte Geometrien, exemplarisch untersucht. Auch der Einfluss von Schwellk{\"a}sten auf die Tonentstehung und die Klangbildung der Orgelpfeife wird studiert. In weiteren, neuartigen Synchronisationsexperimenten mit identisch gestimmten Orgelpfeifen, sowie mit Mixturen wird die Synchronisation f{\"u}r verschiedene, horizontale und vertikale Pfeifenabst{\"a}nde in der Ebene der Schallabstrahlung, untersucht. Die dabei erstmalig beobachteten r{\"a}umlich isotropen Unstetigkeiten im Schwingungsverhalten der gekoppelten Pfeifensysteme, deuten auf abstandsabh{\"a}ngige Wechsel zwischen gegen- und gleichphasigen Sychronisationsregimen hin. Abschließend wird die M{\"o}glichkeit dokumentiert, das Ph{\"a}nomen der Synchronisation zweier Orgelpfeifen durch numerische Simulationen, also der Behandlung der kompressiblen Navier-Stokes Gleichungen mit entsprechenden Rand- und Anfangsbedingungen, realit{\"a}tsnah abzubilden. Auch dies stellt ein Novum dar.}, language = {de} } @misc{Fischer2012, type = {Master Thesis}, author = {Fischer, Jost}, title = {{\"U}ber Synchronisationsph{\"a}nomene nichtlinearer akustischer Oszillatoren}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63618}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {In dieser Arbeit werden die Effekte der Synchronisation nichtlinearer, akustischer Oszillatoren am Beispiel zweier Orgelpfeifen untersucht. Aus vorhandenen, experimentellen Messdaten werden die typischen Merkmale der Synchronisation extrahiert und dargestellt. Es folgt eine detaillierte Analyse der {\"U}bergangsbereiche in das Synchronisationsplateau, der Ph{\"a}nomene w{\"a}hrend der Synchronisation, als auch das Austreten aus der Synchronisationsregion beider Orgelpfeifen, bei verschiedenen Kopplungsst{\"a}rken. Die experimentellen Befunde werfen Fragestellungen nach der Kopplungsfunktion auf. Dazu wird die Tonentstehung in einer Orgelpfeife untersucht. Mit Hilfe von numerischen Simulationen der Tonentstehung wird der Frage nachgegangen, welche fluiddynamischen und aero-akustischen Ursachen die Tonentstehung in der Orgelpfeife hat und inwiefern sich die Mechanismen auf das Modell eines selbsterregten akustischen Oszillators abbilden l{\"a}sst. Mit der Methode des Coarse Graining wird ein Modellansatz formuliert.}, language = {de} } @phdthesis{Massie2011, author = {Massie, Thomas Michael}, title = {Dynamic behavior of phytoplankton populations far from steady state : chemostat experiments and mathematical modeling}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-58102}, school = {Universit{\"a}t Potsdam}, year = {2011}, abstract = {Nature changes continuously and is only seemingly at equilibrium. Environmental parameters like temperature, humidity or insolation may strongly fluctuate on scales ranging from seconds to millions of years. Being part of an ecosystem, species have to cope with these environmental changes. For ecologists, it is of special interest how individual responses to environmental changes affect the dynamics of an entire population - and, if this behavior is predictable. In this context, the demographic structure of a population plays a decisive role since it originates from processes of growth and mortality. These processes are fundamentally influenced by the environment. But, how exactly does the environment influence the behavior of populations? And what does the transient behavior look like? As a result from environmental influences on demography, so called cohorts form. They are age or size classes that are disproportionally represented in the demographic distribution of a population. For instance, if most old and young individuals die due to a cold spell, the population finally consists of mainly middle-aged individuals. Hence, the population got synchronized. Such a population tends to show regular fluctuations in numbers (denoted as oscillations) since the alternating phases of individual growth and population growth (due to reproduction) are now performed synchronously by the majority of the population.That is, one time the population growths, and the other time it declines due to mortality. Synchronous behavior is one of the most pervasive phenomena in nature. Gravitational synchrony in the solar system; fireflies flashing in unison; coordinate firing of pacemaker cells in the heart; electrons in a superconductor marching in lockstep. Whatever scale one looks at, in animate as well as inanimate systems, one is likely to encounter synchrony. In experiments with phytoplankton populations, I could show that this principle of synchrony (as used by physicists) could well-explain the oscillations observed in the experiments, too. The size of the fluctuations depended on the strength by which environmental parameters changed as well as on the demographic state of a population prior to this change. That is, two population living in different habitats can be equally influenced by an environmental change, however, the resulting population dynamics may be significantly different when both populations differed in their demographic state before. Moreover, specific mechanisms relevant for the dynamic behavior of populations, appear only when the environmental conditions change. In my experiments, the population density declined by 50\% after ressource supply was doubled. This counter-intuitive behavior can be explained by increasing ressource consumption. The phytoplankton cells grew larger and enhanced their individual constitution. But at the same time, reproduction was delayed and the population density declined due to the losses by mortality. Environmental influences can also synchronize two or more populations over large distances, which is denoted as Moran effect. Assume two populations living on two distant islands. Although there is no exchange of individuals between them, both populations show a high similarity when comparing their time series. This is because the globally acting climate synchronizes the regionally acting weather on both island. Since the weather fluctuations influence the population dynamics, the Moran effect states that the synchrony between the environment equals the one between the populations. My experiments support this theory and also explain deviations arising when accounting for differences in the populations and the habitats they are living in. Moreover, model simulations and experiments astonishingly show that the synchrony between the populations can be higher than between the environment, when accounting for differences in the environmental fluctuations ("noise color").}, language = {de} } @phdthesis{Bergner2011, author = {Bergner, Andr{\´e}}, title = {Synchronization in complex systems with multiple time scales}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-53407}, school = {Universit{\"a}t Potsdam}, year = {2011}, abstract = {In the present work synchronization phenomena in complex dynamical systems exhibiting multiple time scales have been analyzed. Multiple time scales can be active in different manners. Three different systems have been analyzed with different methods from data analysis. The first system studied is a large heterogenous network of bursting neurons, that is a system with two predominant time scales, the fast firing of action potentials (spikes) and the burst of repetitive spikes followed by a quiescent phase. This system has been integrated numerically and analyzed with methods based on recurrence in phase space. An interesting result are the different transitions to synchrony found in the two distinct time scales. Moreover, an anomalous synchronization effect can be observed in the fast time scale, i.e. there is range of the coupling strength where desynchronization occurs. The second system analyzed, numerically as well as experimentally, is a pair of coupled CO₂ lasers in a chaotic bursting regime. This system is interesting due to its similarity with epidemic models. We explain the bursts by different time scales generated from unstable periodic orbits embedded in the chaotic attractor and perform a synchronization analysis of these different orbits utilizing the continuous wavelet transform. We find a diverse route to synchrony of these different observed time scales. The last system studied is a small network motif of limit cycle oscillators. Precisely, we have studied a hub motif, which serves as elementary building block for scale-free networks, a type of network found in many real world applications. These hubs are of special importance for communication and information transfer in complex networks. Here, a detailed study on the mechanism of synchronization in oscillatory networks with a broad frequency distribution has been carried out. In particular, we find a remote synchronization of nodes in the network which are not directly coupled. We also explain the responsible mechanism and its limitations and constraints. Further we derive an analytic expression for it and show that information transmission in pure phase oscillators, such as the Kuramoto type, is limited. In addition to the numerical and analytic analysis an experiment consisting of electrical circuits has been designed. The obtained results confirm the former findings.}, 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{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} } @phdthesis{PereiradaSilva2007, author = {Pereira da Silva, Tiago}, title = {Synchronization in active networks}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14347}, school = {Universit{\"a}t Potsdam}, year = {2007}, abstract = {In nature one commonly finds interacting complex oscillators which by the coupling scheme form small and large networks, e.g. neural networks. Surprisingly, the oscillators can synchronize, still preserving the complex behavior. Synchronization is a fundamental phenomenon in coupled nonlinear oscillators. Synchronization can be enhanced at different levels, that is, the constraints on which the synchronization appears. Those can be in the trajectory amplitude, requiring the amplitudes of both oscillators to be equal, giving place to complete synchronization. Conversely, the constraint could also be in a function of the trajectory, e.g. the phase, giving place to phase synchronization (PS). In this case, one requires the phase difference between both oscillators to be finite for all times, while the trajectory amplitude may be uncorrelated. The study of PS has shown its relevance to important technological problems, e.g. communication, collective behavior in neural networks, pattern formation, Parkinson disease, epilepsy, as well as behavioral activities. It has been reported that it mediates processes of information transmission and collective behavior in neural and active networks and communication processes in the Human brain. In this work, we have pursed a general way to analyze the onset of PS in small and large networks. Firstly, we have analyzed many phase coordinates for compact attractors. We have shown that for a broad class of attractors the PS phenomenon is invariant under the phase definition. Our method enables to state about the existence of phase synchronization in coupled chaotic oscillators without having to measure the phase. This is done by observing the oscillators at special times, and analyzing whether this set of points is localized. We have show that this approach is fruitful to analyze the onset of phase synchronization in chaotic attractors whose phases are not well defined, as well as, in networks of non-identical spiking/bursting neurons connected by chemical synapses. Moreover, we have also related the synchronization and the information transmission through the conditional observations. In particular, we have found that inside a network clusters may appear. These can be used to transmit more than one information, which provides a multi-processing of information. Furthermore, These clusters provide a multichannel communication, that is, one can integrate a large number of neurons into a single communication system, and information can arrive simultaneously at different places of the network.}, 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 = { 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{Bergweiler2005, author = {Bergweiler, Steffen}, title = {K{\"o}rperoszillation und Schallabstrahlung akustischer Wellenleiter unter Ber{\"u}cksichtigung von Wandungseinfl{\"u}ssen und Kopplungseffekten : Ver{\"a}ndern Metalllegierung und Wandungsprofil des Rohrresonators den Klang der labialen Orgelpfeife?}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6561}, school = {Universit{\"a}t Potsdam}, year = {2005}, abstract = {Am Beispiel der Orgelpfeife wurde der Einfluss der Wandungsgeometrie des akustischen Wellenleiters auf die Schallabstrahlung untersucht. F{\"u}r verschiedene Metalllegierungen wurden unterschiedliche Profile der Orgelpfeifenwandung verglichen: ein konisches Wandungsprofil mit zur M{\"u}ndung hin abnehmender Wandungsst{\"a}rke und ein paralleles Wandungsprofil mit konstanter Wandungsst{\"a}rke. F{\"u}r eine hohe statistische Sicherheit der Ergebnisse wurden s{\"a}mtliche Untersuchungen an vier mal zehn Testpfeifen durchgef{\"u}hrt. Mit Ausnahme der beschriebenen Unterschiede sind die Pfeifen von gleichen Abmessungen und auf gleichen Klang intoniert. Die {\"U}berpr{\"u}fung der Wandungseinfl{\"u}sse auf den Klang besteht aus drei verschiedenen Untersuchungen: Erstens, einer subjektiven Hinterfragung der Wahrnehmbarkeit in einem H{\"o}rtest. Zweitens wurde der abgestrahlte Luftschall objektiv gemessen und das Spektrum der Pfeifen in seinen Komponenten (Teilt{\"o}ne, Grundfrequenz) verglichen. Drittens wurde mit einer neuartigen Messtechnik die Oszillation des Pfeifenk{\"o}rpers (ein einem akustischen Monopol entsprechendes "Atmen" des Querschnitts) untersucht. Die Ergebnisse belegen die Wahrnehmbarkeit unterschiedlicher Wandungsprofile als auch klare objektive Differenzen zwischen den emittierten Schallspektren. Ein Atmen mit guter Korrelation zur inneren Druckanregung best{\"a}tigt den Einfluss wandungsprofilabh{\"a}ngiger Oszillationen auf den Klang der Orgelpfeife. Schließlich wurde die Interaktion zweier in Abstand und Grundfrequenz nah beieinander liegender Orgelpfeifen {\"u}berpr{\"u}ft. Als Ursache des dabei wahrnehmbaren Oktavsprung des Orgeltons konnte eine gegenphasiger Oszillation des Grundtons beider Pfeifen nachgewiesen werden.}, subject = {Schallabstrahlung}, language = {de} } @phdthesis{Allefeld2004, author = {Allefeld, Carsten}, title = {Phase synchronization analysis of event-related brain potentials in language processing}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001873}, school = {Universit{\"a}t Potsdam}, year = {2004}, abstract = {Das Forschungsthema Synchronisation bildet einen Schnittpunkt von Nichtlinearer Dynamik und Neurowissenschaft. So hat zum einen neurobiologische Forschung gezeigt, daß die Synchronisation neuronaler Aktivit{\"a}t einen wesentlichen Aspekt der Funktionsweise des Gehirns darstellt. Zum anderen haben Fortschritte in der physikalischen Theorie zur Entdeckung des Ph{\"a}nomens der Phasensynchronisation gef{\"u}hrt. Eine dadurch motivierte Datenanalysemethode, die Phasensynchronisations-Analyse, ist bereits mit Erfolg auf empirische Daten angewandt worden. Die vorliegende Dissertation kn{\"u}pft an diese konvergierenden Forschungslinien an. Ihren Gegenstand bilden methodische Beitr{\"a}ge zur Fortentwicklung der Phasensynchronisations-Analyse, sowie deren Anwendung auf ereigniskorrelierte Potentiale, eine besonders in den Kognitionswissenschaften wichtige Form von EEG-Daten. Die methodischen Beitr{\"a}ge dieser Arbeit bestehen zum ersten in einer Reihe spezialisierter statistischer Tests auf einen Unterschied der Synchronisationsst{\"a}rke in zwei verschiedenen Zust{\"a}nden eines Systems zweier Oszillatoren. Zweitens wird im Hinblick auf den viel-kanaligen Charakter von EEG-Daten ein Ansatz zur multivariaten Phasensynchronisations-Analyse vorgestellt. Zur empirischen Untersuchung neuronaler Synchronisation wurde ein klassisches Experiment zur Sprachverarbeitung repliziert, in dem der Effekt einer semantischen Verletzung im Satzkontext mit demjenigen der Manipulation physischer Reizeigenschaften (Schriftfarbe) verglichen wird. Hier zeigt die Phasensynchronisations-Analyse eine Verringerung der globalen Synchronisationsst{\"a}rke f{\"u}r die semantische Verletzung sowie eine Verst{\"a}rkung f{\"u}r die physische Manipulation. Im zweiten Fall l{\"a}ßt sich der global beobachtete Synchronisationseffekt mittels der multivariaten Analyse auf die Interaktion zweier symmetrisch gelegener Gehirnareale zur{\"u}ckf{\"u}hren. Die vorgelegten Befunde zeigen, daß die physikalisch motivierte Methode der Phasensynchronisations-Analyse einen wesentlichen Beitrag zur Untersuchung ereigniskorrelierter Potentiale in den Kognitionswissenschaften zu leisten vermag.}, 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{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{Ahlers2001, author = {Ahlers, Volker}, title = {Scaling and synchronization in deterministic and stochastic nonlinear dynamical systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0000320}, school = {Universit{\"a}t Potsdam}, year = {2001}, abstract = {Gegenstand dieser Arbeit ist die Untersuchung universeller Skalengesetze, die in gekoppelten chaotischen Systemen beobachtet werden. Ergebnisse werden erzielt durch das Ersetzen der chaotischen Fluktuationen in der St{\"o}rungsdynamik durch stochastische Prozesse. Zun{\"a}chst wird ein zeitkontinuierliches stochastisches Modell f{\"u}rschwach gekoppelte chaotische Systeme eingef{\"u}hrt, um die Skalierung der Lyapunov-Exponenten mit der Kopplungsst{\"a}rke (coupling sensitivity of chaos) zu untersuchen. Mit Hilfe der Fokker-Planck-Gleichung werden Skalengesetze hergeleitet, die von Ergebnissen numerischer Simulationen best{\"a}tigt werden. Anschließend wird der neuartige Effekt der vermiedenen Kreuzung von Lyapunov-Exponenten schwach gekoppelter ungeordneter chaotischer Systeme beschrieben, der qualitativ der Abstoßung zwischen Energieniveaus in Quantensystemen {\"a}hnelt. Unter Benutzung der f{\"u}r die coupling sensitivity of chaos gewonnenen Skalengesetze wird ein asymptotischer Ausdruck f{\"u}r die Verteilungsfunktion kleiner Abst{\"a}nde zwischen Lyapunov-Exponenten hergeleitet und mit Ergebnissen numerischer Simulationen verglichen. Schließlich wird gezeigt, dass der Synchronisations{\"u}bergang in starkgekoppelten r{\"a}umlich ausgedehnten chaotischen Systemen einem kontinuierlichen Phasen{\"u}bergang entspricht, mit der Kopplungsst{\"a}rke und dem Synchronisationsfehler als Kontroll- beziehungsweise Ordnungsparameter. Unter Benutzung von Ergebnissen numerischer Simulationen sowie theoretischen {\"U}berlegungen anhand einer partiellen Differentialgleichung mit multiplikativem Rauschen werden die Universalit{\"a}tsklassen der zwei beobachteten {\"U}bergangsarten bestimmt (Kardar-Parisi-Zhang-Gleichung mit S{\"a}ttigungsterm, gerichtete Perkolation).}, language = {en} }