1,150 research outputs found
Comment on "Long Time Evolution of Phase Oscillator Systems" [Chaos 19,023117 (2009), arXiv:0902.2773]
A previous paper (arXiv:0902.2773, henceforth referred to as I) considered a
general class of problems involving the evolution of large systems of globally
coupled phase oscillators. It was shown there that, in an appropriate sense,
the solutions to these problems are time asymptotically attracted toward a
reduced manifold of system states (denoted M). This result has considerable
utility in the analysis of these systems, as has been amply demonstrated in
recent papers. In this note, we show that the analysis of I can be modified in
a simple way that establishes significant extensions of the range of validity
of our previous result. In particular, we generalize I in the following ways:
(1) attraction to M is now shown for a very general class of oscillator
frequency distribution functions g(\omega), and (2) a previous restriction on
the allowed class of initial conditions is now substantially relaxed
Synchronization in large directed networks of coupled phase oscillators
We extend recent theoretical approximations describing the transition to
synchronization in large undirected networks of coupled phase oscillators to
the case of directed networks. We also consider extensions to networks with
mixed positive/negative coupling strengths. We compare our theory with
numerical simulations and find good agreement
Approximating the largest eigenvalue of network adjacency matrices
The largest eigenvalue of the adjacency matrix of a network plays an
important role in several network processes (e.g., synchronization of
oscillators, percolation on directed networks, linear stability of equilibria
of network coupled systems, etc.). In this paper we develop approximations to
the largest eigenvalue of adjacency matrices and discuss the relationships
between these approximations. Numerical experiments on simulated networks are
used to test our results.Comment: 7 pages, 4 figure
The emergence of coherence in complex networks of heterogeneous dynamical systems
We present a general theory for the onset of coherence in collections of
heterogeneous maps interacting via a complex connection network. Our method
allows the dynamics of the individual uncoupled systems to be either chaotic or
periodic, and applies generally to networks for which the number of connections
per node is large. We find that the critical coupling strength at which a
transition to synchrony takes place depends separately on the dynamics of the
individual uncoupled systems and on the largest eigenvalue of the adjacency
matrix of the coupling network. Our theory directly generalizes the Kuramoto
model of equal strength, all-to-all coupled phase oscillators to the case of
oscillators with more realistic dynamics coupled via a large heterogeneous
network.Comment: 4 pages, 1 figure. Published versio
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