6,300 research outputs found
Influence of cumulative damage on synchronization of Kuramoto oscillators on networks
In this paper, we study the synchronization of identical Kuramoto phase
oscillators under cumulative stochastic damage to the edges of networks. We
analyze the capacity of coupled oscillators to reach a coherent state from
initial random phases. The process of synchronization is a global function
performed by a system that gradually changes when the damage weakens individual
connections of the network. We explore diverse structures characterized by
different topologies. Among these are deterministic networks as a wheel or the
lattice formed by the movements of the knight on a chess board, and random
networks generated with the Erd\H{o}s-R\'enyi and Barab\'asi-Albert algorithms.
In addition, we study the synchronization times of 109 non-isomorphic graphs
with six nodes. The synchronization times and other introduced quantities are
sensitive to the impact of damage, allowing us to measure the reduction of the
capacity of synchronization and classify the effect of damage in the systems
under study. This approach is general and paves the way for the exploration of
the effect of damage accumulation in diverse dynamical processes in complex
systems.Comment: 27 pages, 9 figure
Synchronization in complex networks
Synchronization processes in populations of locally interacting elements are
in the focus of intense research in physical, biological, chemical,
technological and social systems. The many efforts devoted to understand
synchronization phenomena in natural systems take now advantage of the recent
theory of complex networks. In this review, we report the advances in the
comprehension of synchronization phenomena when oscillating elements are
constrained to interact in a complex network topology. We also overview the new
emergent features coming out from the interplay between the structure and the
function of the underlying pattern of connections. Extensive numerical work as
well as analytical approaches to the problem are presented. Finally, we review
several applications of synchronization in complex networks to different
disciplines: biological systems and neuroscience, engineering and computer
science, and economy and social sciences.Comment: Final version published in Physics Reports. More information
available at http://synchronets.googlepages.com
Dynamical and spectral properties of complex networks
Dynamical properties of complex networks are related to the spectral
properties of the Laplacian matrix that describes the pattern of connectivity
of the network. In particular we compute the synchronization time for different
types of networks and different dynamics. We show that the main dependence of
the synchronization time is on the smallest nonzero eigenvalue of the Laplacian
matrix, in contrast to other proposals in terms of the spectrum of the
adjacency matrix. Then, this topological property becomes the most relevant for
the dynamics.Comment: 14 pages, 5 figures, to be published in New Journal of Physic
Symbolic Synchronization and the Detection of Global Properties of Coupled Dynamics from Local Information
We study coupled dynamics on networks using symbolic dynamics. The symbolic
dynamics is defined by dividing the state space into a small number of regions
(typically 2), and considering the relative frequencies of the transitions
between those regions. It turns out that the global qualitative properties of
the coupled dynamics can be classified into three different phases based on the
synchronization of the variables and the homogeneity of the symbolic dynamics.
Of particular interest is the {\it homogeneous unsynchronized phase} where the
coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)
identical symbolic dynamics at all the nodes in the network. We refer to this
dynamical behaviour as {\it symbolic synchronization}. In this phase, the local
symbolic dynamics of any arbitrarily selected node reflects global properties
of the coupled dynamics, such as qualitative behaviour of the largest Lyapunov
exponent and phase synchronization. This phase depends mainly on the network
architecture, and only to a smaller extent on the local chaotic dynamical
function. We present results for two model dynamics, iterations of the
one-dimensional logistic map and the two-dimensional H\'enon map, as local
dynamical function.Comment: 21 pages, 7 figure
- …