Collective Dynamics in Complex Networks of Noisy Phase Oscillators
Bernard Sonnenschein
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Bernard Sonnenschein, Collective Dynamics in Complex Networks of Noisy Phase Oscillators (2016), Logos Verlag, Berlin, ISBN: 9783832588250
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Beschreibung / Abstract
This work aims to contribute to our understanding of the effects of noise and non-uniform interactions in populations of oscillatory units. In particular, we explore the collective dynamics in various extensions of the Kuramoto model. We develop a theoretical framework to study such noisy systems and we show through many examples that indeed new insights can be gained with our method. The first step is to coarse-grain the complex networks. The oscillatory units are then characterized solely by their individual quantities, so that identical units can be grouped together. The second step consists of the ansatz that in all these groups the distributions of the oscillators' phases follow time-dependent Gaussians. We apply this analytical two-step method to oscillator networks with correlations between coupling strengths and natural frequencies, to populations with mixed positive and negative coupling strengths, and to noise-driven active rotators, which can perform excitable dynamics. We calculate the rich phase diagrams that delineate the emergent rhythms. Extensive numerical simulations are performed to show both the validity and the limitations of our theoretical results.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 2 Onset of synchronization in networks of noisy phase oscillators
- 2.1 Coarse-graining uncorrelated random networks
- 2.2 The nonlinear Fokker-Planck equation
- 2.3 The critical coupling strength
- 2.4 Application to dense small-world networks
- 2.5 Interpolating between sparse and dense networks
- 2.6 Networks with degree-dependent frequency dispersion
- 2.7 Implementation of an example with first numerical experiments
- 2.8 Detailed analysis and clarification of noise effects
- 2.9 Summary and outlook
- 3 Approximate solution to the stochastic Kuramoto model
- 3.1 Exposition of the Gaussian approximation
- 3.2 Time-dependent solutions and long-time limits
- 3.3 Gaussian theory vs. numerical experiments and exact results
- 3.4 Temporal fluctuations vs. quenched disorder
- 3.5 Including complex networks
- 3.6 Summary and outlook
- 4 Excitable elements controlled by noise and network structure
- 4.1 Noise-driven active rotators
- 4.2 First example: regular networks
- 4.3 Second example: binary random networks
- 4.4 Cooperative behavior between self-oscillatory and excitable units
- 4.5 Bifurcation diagrams and order parameters for a concrete example
- 4.6 Summary and outlook
- 5 Noisy oscillators with asymmetric attractive-repulsive interactions
- 5.1 Bifurcation diagrams
- 5.2 Simulation results
- 5.3 Towards an arbitrary number of populations
- 5.4 Summary and outlook
- 6 Synchronization in the stochastic Kuramoto-Sakaguchi model
- 6.1 Application of Gaussian approximation
- 6.2 Connection between common frequency and synchronization
- 6.3 Towards a new phenomenological theory
- 6.4 Summary and outlook
- 7 Conclusions
- A Directed networks
- B Beyond uncorrelated networks
- B.1 Two-point correlated random networks
- B.2 Assortativity by degree
- B.3 Assortativity by frequency
- Bibliography