TY - JOUR A1 - Zheng, Chunming A1 - Tönjes, Ralf A1 - Pikovskij, Arkadij T1 - Transition to synchrony in a three-dimensional swarming model with helical trajectories JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found that alignment of the velocities occurs discontinuously for arbitrarily small attractive coupling. Adding noise to the system resolves this singular limit and leads to a continuous transition, either to a directed collective motion or to center-of-mass rotations. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.014216 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Zheng, Chunming A1 - Tönjes, Ralf T1 - Noise-induced swarming of active particles JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We report on the effect of spatially correlated noise on the velocities of self-propelled particles. Correlations in the random forces acting on self-propelled particles can induce directed collective motion, i.e., swarming. Even with repulsive coupling in the velocity directions, which favors a disordered state, strong correlations in the fluctuations can align the velocities locally leading to a macroscopic, turbulent velocity field. On the other hand, while spatially correlated noise is aligning the velocities locally, the swarming transition to globally directed motion is inhibited when the correlation length of the noise is nonzero, but smaller than the system size. We analyze the swarming transition in d-dimensional space in a mean field model of globally coupled velocity vectors. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.106.064601 SN - 2470-0045 SN - 2470-0053 VL - 106 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Tönjes, Ralf A1 - Sokolov, Igor M. A1 - Postnikov, Eugene B. T1 - Spectral properties of the fractional Fokker-Planck operator for the Levy flight in a harmonic potential JF - The European physical journal N2 - We present a detailed analysis of the eigenfunctions of the Fokker-Planck operator for the LevyOrnstein- Uhlenbeck process, their asymptotic behavior and recurrence relations, explicit expressions in coordinate space for the special cases of the Ornstein-Uhlenbeck process with Gaussian and with Cauchy white noise and for the transformation kernel, which maps the fractional Fokker-Planck operator of the Cauchy-Ornstein-Uhlenbeck process to the non-fractional Fokker-Planck operator of the usual Gaussian Ornstein-Uhlenbeck process. We also describe how non-spectral relaxation can be observed in bounded random variables of the Levy-Ornstein-Uhlenbeck process and their correlation functions. Y1 - 2014 U6 - https://doi.org/10.1140/epjb/e2014-50558-5 SN - 1434-6028 SN - 1434-6036 VL - 87 IS - 12 PB - Springer CY - New York ER - TY - JOUR A1 - Tönjes, Ralf A1 - Sokolov, Igor M. A1 - Postnikov, E. B. T1 - Non-spectral relaxation in one dimensional Ornstein-Uhlenbeck processes JF - Physical review letters N2 - The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which growat infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Levy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Levy noise, such initial conditions are the rule. DOI: 10.1103/PhysRevLett.110.150602 Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevLett.110.150602 SN - 0031-9007 VL - 110 IS - 15 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Tönjes, Ralf A1 - Pikovsky, Arkady T1 - Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study ensembles of globally coupled or forced identical phase oscillators subject to independent white Cauchy noise. We demonstrate that if the oscillators are forced in several harmonics, stationary synchronous regimes can be exactly described with a finite number of complex order parameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions. For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.052315 SN - 2470-0045 SN - 2470-0053 VL - 102 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Tönjes, Ralf A1 - Kori, Hiroshi T1 - Phase and frequency linear response theory for hyperbolic chaotic oscillators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response. Y1 - 2022 U6 - https://doi.org/10.1063/5.0064519 SN - 1054-1500 SN - 1089-7682 VL - 32 IS - 4 PB - AIP Publishing CY - Melville ER - TY - JOUR A1 - Tönjes, Ralf A1 - Fiore, Carlos E. A1 - Pereira da Silva, Tiago T1 - Coherence resonance in influencer networks JF - Nature Communications N2 - Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite the abundance of networks with influencers, little is known about their response to stochastic forcing. Here, for oscillatory dynamics on influencer networks, we show that subjecting influencers to an optimal intensity of noise can result in enhanced network synchronization. This new network dynamical effect, which we call coherence resonance in influencer networks, emerges from a synergy between network structure and stochasticity and is highly nonlinear, vanishing when the noise is too weak or too strong. Our results reveal that the influencer backbone can sharply increase the dynamical response in complex systems of coupled oscillators. Influencer networks include a small set of highly-connected nodes and can reach synchrony only via strong node interaction. Tonjes et al. show that introducing an optimal amount of noise enhances synchronization of such networks, which may be relevant for neuroscience or opinion dynamics applications. Y1 - 2021 U6 - https://doi.org/10.1038/s41467-020-20441-4 SN - 2041-1723 VL - 12 IS - 1 PB - Nature Publishing Group UK CY - London ER - TY - JOUR A1 - Tönjes, Ralf A1 - Blasius, Bernd T1 - Perturbation analysis of the Kuramoto phase-diffusion equation subject to quenched frequency disorder N2 - The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point. Y1 - 2009 UR - http://pre.aps.org/ U6 - https://doi.org/10.1103/Physreve.79.016112 SN - 1539-3755 ER - TY - JOUR A1 - Tönjes, Ralf A1 - Blasius, Bernd T1 - Perturbation analysis of complete synchronization in networks of phase oscillators N2 - The behavior of weakly coupled self-sustained oscillators can often be well described by phase equations. Here we use the paradigm of Kuramoto phase oscillators which are coupled in a network to calculate first- and second-order corrections to the frequency of the fully synchronized state for nonidentical oscillators. The topology of the underlying coupling network is reflected in the eigenvalues and eigenvectors of the network Laplacian which influence the synchronization frequency in a particular way. They characterize the importance of nodes in a network and the relations between them. Expected values for the synchronization frequency are obtained for oscillators with quenched random frequencies on a class of scale-free random networks and for a Erdoumls-Reacutenyi random network. We briefly discuss an application of the perturbation theory in the second order to network structural analysis. Y1 - 2009 UR - http://pre.aps.org/ U6 - https://doi.org/10.1103/Physreve.80.026202 SN - 1539-3755 ER - TY - THES A1 - Tönjes, Ralf T1 - Pattern formation through synchronization in systems of nonidentical autonomous oscillators T1 - Musterbildung durch Synchronisation in Systemen nicht identischer, autonomer Oszillatoren N2 - 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. N2 - Die vorliegende Arbeit beschäftigt sich in Theorie und Simulation mit den raum-zeitlichen Strukturen, die entstehen, wenn nicht-identische, diffusiv gekoppelte Oszillatoren synchronisieren. Wir greifen dabei auf die von Kuramoto hergeleiteten Phasengleichungen zurück. Eine entscheidene Rolle für die Musterbildung spielt die Symmetrie der Kopplung. Wir untersuchen den ordnenden Einfluss von Asymmetrie (Nichtisochronizität) in der Phasenkopplungsfunktion auf das Phasenprofil in Synchronisation und das Zusammenspiel zwischen dieser Asymmetrie und der Frequenzheterogenität im System. Die Arbeit gliedert sich in drei Hauptteile. Kapitel 2 und 3 beschäftigen sich mit den grundlegenden Gleichungen und den Bedingungen für stabile Synchronisation. Im Kapitel 4 charakterisieren wir die Phasenprofile in Synchronisation für verschiedene Spezialfälle sowie in der von uns eingeführten exponentiellen Approximation der Phasenkopplungsfunktion. Schliesslich untersuchen wir im dritten Teil (Kap.5) den Einfluss von Nichtisochronizität auf die Synchronisationsfrequenz in kontinuierlichen, oszillatorischen Reaktions-Diffusionssystemen und diskreten Netzwerken von Oszillatoren. KW - Synchronisation KW - Musterbildung KW - Phasen-Gleichungen KW - Phasen-Oszillatoren KW - Kuramoto Modell KW - synchronization KW - pattern formation KW - phase equations KW - phase oscillators KW - Kuramoto model Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15973 ER - TY - THES A1 - Tönjes, Ralf T1 - On the effects of disorder on the ability of oscillatory or directional dynamics to synchronize N2 - In this thesis I present a collection of publications of my work, containing analytic results and observations in numerical experiments on the effects of various inhomogeneities, on the ability of coupled oscillators to synchronize their collective dynamics. Most of these works are concerned with the effects of Gaussian and non-Gaussian noise acting on the phase of autonomous oscillators (Secs. 2.1-2.4) or on the direction of higher dimensional state vectors (Secs. 2.5,2.6). I obtain exact and approximate solutions to the non-linear equations governing the distributions of phases, or perform linear stability analysis of the uniform distribution to obtain the transition point from a completely disordered state to partial order or more complicated collective behavior. Other inhomogeneities, that can affect synchronization of coupled oscillators, are irregular, chaotic oscillations or a complex, and possibly random structure in the coupling network. In Section 2.9 I present a new method to define the phase- and frequency linear response function for chaotic oscillators. In Sections 2.4, 2.7 and 2.8 I study synchronization in complex networks of coupled oscillators. Each section in Chapter 2 - Manuscripts, is devoted to one research paper and begins with a list of the main results, a description of my contributions to the work and a short account of the scientific context, i.e. the questions and challenges which started the research and the relation of the work to my other research projects. The manuscripts in this thesis are reproductions of the arXiv versions, i.e. preprints under the creative commons licence. N2 - In dieser Habilitationsschrift präsentiere ich eine Sammlung von Veröffentlichungen meiner Arbeit, die analytische Ergebnisse und Beobachtungen aus numerischen Experimenten zu den Effekten verschiedener Inhomogenitäten auf die Fähigkeit gekoppelter Oszillatoren zur Synchronisation ihrer kollektiven Dynamik enthält. Die meisten dieser Arbeiten befassen sich mit den Effekten von gaußschem und nicht-gaußschem Rauschen, das auf die Phasen autonomer Oszillatoren einwirkt (Abschnitte 2.1-2.4) oder auf die Richtung von höherdimensionalen Zustandsvektoren (Abschnitte 2.5, 2.6). Ich erhalte exakte und approximative Lösungen für die nichtlinearen Gleichungen, die die Verteilung der Phasen bestimmen, oder führe eine lineare Stabilitätsanalyse der Gleichverteilung durch, um den Übergangspunkt von einem vollständig ungeordneten Zustand zu partieller Ordnung oder komplexerem kollektiven Verhalten zu ermitteln. Andere Inhomogenitäten, die die Synchronisation gekoppelter Oszillatoren beeinflussen können, sind unregelmäßige, chaotische Oszillationen oder eine komplexe und möglicherweise zufällige Struktur im Kopplungsnetzwerk. In Abschnitt 2.9 präsentiere ich eine neue Methode zur Definition der Phasen- und Frequenzantwortfunktion für chaotische Oszillatoren. In den Abschnitten 2.4, 2.7 und 2.8 untersuche ich die Synchronisation in komplexen Netzwerken gekoppelter Oszillatoren. Jeder Abschnitt in Kapitel 2 - Manuskripte, ist einer Forschungsarbeit gewidmet und beginnt mit einer Liste der wichtigsten Ergebnisse, einer Beschreibung meiner Beiträge zur Arbeit und einem kurzen Überblick über den wissenschaftlichen Kontext, d.h. die Fragen und Herausforderungen, die die Forschung ausgelöst haben, sowie der Zusammenhang der Arbeit mit meinen anderen Forschungsprojekten. Die Manuskripte in dieser Dissertation sind Nachdrucke der arXiv-Versionen, d.h. Vorabdrucke unter der Creative Commons Lizenz. T2 - Über die Wirkung von Unordnung auf die Synchronisierbarkeit von oszillatorischer oder gerichteter Dynamik KW - synchronization KW - oscillators KW - dynamics on networks KW - Dynamik in Netzwerken KW - Oszillatoren KW - Synchronisation Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-651942 ER - TY - JOUR A1 - Sebek, Michael A1 - Tönjes, Ralf A1 - Kiss, Istvan Z. T1 - Complex Rotating Waves and Long Transients in a Ring Network of Electrochemical Oscillators with Sparse Random Cross-Connections JF - Physical review letters N2 - We perform experiments and phase model simulations with a ring network of oscillatory electrochemical reactions to explore the effect of random connections and nonisochronicity of the interactions on the pattern formation. A few additional links facilitate the emergence of the fully synchronized state. With larger nonisochronicity, complex rotating waves or persistent irregular phase dynamics can derail the convergence to global synchronization. The observed long transients of irregular phase dynamics exemplify the possibility of a sudden onset of hypersynchronous behavior without any external stimulus or network reorganization. Y1 - 2016 U6 - https://doi.org/10.1103/PhysRevLett.116.068701 SN - 0031-9007 SN - 1079-7114 VL - 116 SP - 3001 EP - 3009 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Gong, Chen Chris A1 - Zheng, Chunming A1 - Tönjes, Ralf A1 - Pikovskij, Arkadij T1 - Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. Published under license by AIP Publishing. Y1 - 2019 U6 - https://doi.org/10.1063/1.5084144 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 3 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Gong, Chen Chris A1 - Tönjes, Ralf A1 - Pikovsky, Arkady T1 - Coupled Möbius maps as a tool to model Kuramoto phase synchronization JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We propose Mobius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of nonidentical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and differences between the iterated map models and their known continuous-time counterparts. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.022206 SN - 2470-0045 SN - 2470-0053 SN - 1063-651X SN - 2470-0061 SN - 1550-2376 VL - 102 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Blasius, Bernd A1 - Tönjes, Ralf T1 - Quasiregular concentric waves in heterogeneous lattices of coupled oscillators N2 - We study the pattern formation in a lattice of locally coupled phase oscillators with quenched disorder. In the synchronized regime quasi regular concentric waves can arise which are induced by the disorder of the system. Maximal regularity is found at the edge of the synchronization regime. The emergence of the concentric waves is related to the symmetry breaking of the interaction function. An explanation of the numerically observed phenomena is given in a one- dimensional chain of coupled phase oscillators. Scaling properties, describing the target patterns are obtained. Y1 - 2005 UR - http://www.agnld.uni-potsdam.de/~bernd/papers/prl1.pdf ER - TY - JOUR A1 - Blasius, Bernd A1 - Tönjes, Ralf T1 - Zipf's Law in the popularity distribution of chess openings N2 - We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power law with an exponent that increases linearly with the game depth, whereas the pooled distribution of all opening weights follows Zipf's law with universal exponent. We propose a simple stochastic process that is able to capture the observed playing statistics and show that the Zipf law arises from the self-similar nature of the game tree of chess. Thus, in the case of hierarchical fragmentation the scaling is truly universal and independent of a particular generating mechanism. Our findings are of relevance in general processes with composite decisions. Y1 - 2009 UR - http://prl.aps.org/ U6 - https://doi.org/10.1103/Physrevlett.103.218701 SN - 0031-9007 ER -