@article{ToenjesFiorePereiradaSilva2021, author = {T{\"o}njes, Ralf and Fiore, Carlos E. and Pereira da Silva, Tiago}, title = {Coherence resonance in influencer networks}, series = {Nature Communications}, volume = {12}, journal = {Nature Communications}, number = {1}, publisher = {Nature Publishing Group UK}, address = {London}, issn = {2041-1723}, doi = {10.1038/s41467-020-20441-4}, pages = {8}, year = {2021}, abstract = {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.}, language = {en} } @article{SebekToenjesKiss2016, author = {Sebek, Michael and T{\"o}njes, Ralf and Kiss, Istvan Z.}, title = {Complex Rotating Waves and Long Transients in a Ring Network of Electrochemical Oscillators with Sparse Random Cross-Connections}, series = {Physical review letters}, volume = {116}, journal = {Physical review letters}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.116.068701}, pages = {3001 -- 3009}, year = {2016}, abstract = {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.}, language = {en} } @article{GongToenjesPikovsky2020, author = {Gong, Chen Chris and T{\"o}njes, Ralf and Pikovsky, Arkady}, title = {Coupled M{\"o}bius maps as a tool to model Kuramoto phase synchronization}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.022206}, pages = {12}, year = {2020}, abstract = {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.}, language = {en} } @article{ToenjesPikovsky2020, author = {T{\"o}njes, Ralf and Pikovsky, Arkady}, title = {Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.052315}, pages = {5}, year = {2020}, abstract = {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.}, language = {en} } @article{ZhengToenjes2022, author = {Zheng, Chunming and T{\"o}njes, Ralf}, title = {Noise-induced swarming of active particles}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {106}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.106.064601}, pages = {7}, year = {2022}, abstract = {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.}, language = {en} } @article{ToenjesSokolovPostnikov2013, author = {T{\"o}njes, Ralf and Sokolov, Igor M. and Postnikov, E. B.}, title = {Non-spectral relaxation in one dimensional Ornstein-Uhlenbeck processes}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {15}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.150602}, pages = {4}, year = {2013}, abstract = {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}, language = {en} } @phdthesis{Toenjes2024, author = {T{\"o}njes, Ralf}, title = {On the effects of disorder on the ability of oscillatory or directional dynamics to synchronize}, doi = {10.25932/publishup-65194}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-651942}, school = {Universit{\"a}t Potsdam}, pages = {viii, 124}, year = {2024}, abstract = {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.}, 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} } @article{ToenjesBlasius2009, author = {T{\"o}njes, Ralf and Blasius, Bernd}, title = {Perturbation analysis of complete synchronization in networks of phase oscillators}, issn = {1539-3755}, doi = {10.1103/Physreve.80.026202}, year = {2009}, abstract = {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.}, language = {en} } @article{ToenjesBlasius2009, author = {T{\"o}njes, Ralf and Blasius, Bernd}, title = {Perturbation analysis of the Kuramoto phase-diffusion equation subject to quenched frequency disorder}, issn = {1539-3755}, doi = {10.1103/Physreve.79.016112}, year = {2009}, abstract = {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.}, language = {en} }