@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{ToenjesKori2022,
  author    = {T{\"o}njes, Ralf and Kori, Hiroshi},
  title     = {Phase and frequency linear response theory for hyperbolic chaotic oscillators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {32},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {4},
  publisher = {AIP Publishing},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/5.0064519},
  pages     = {13},
  year      = {2022},
  abstract  = {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.},
  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{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}
}