714 research outputs found
Graph partitions and cluster synchronization in networks of oscillators
Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators
Graph partitions and cluster synchronization in networks of oscillators
Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators
Synchronization Patterns in Networks of Kuramoto Oscillators: A Geometric Approach for Analysis and Control
Synchronization is crucial for the correct functionality of many natural and
man-made complex systems. In this work we characterize the formation of
synchronization patterns in networks of Kuramoto oscillators. Specifically, we
reveal conditions on the network weights and structure and on the oscillators'
natural frequencies that allow the phases of a group of oscillators to evolve
cohesively, yet independently from the phases of oscillators in different
clusters. Our conditions are applicable to general directed and weighted
networks of heterogeneous oscillators. Surprisingly, although the oscillators
exhibit nonlinear dynamics, our approach relies entirely on tools from linear
algebra and graph theory. Further, we develop a control mechanism to determine
the smallest (as measured by the Frobenius norm) network perturbation to ensure
the formation of a desired synchronization pattern. Our procedure allows us to
constrain the set of edges that can be modified, thus enforcing the sparsity
structure of the network perturbation. The results are validated through a set
of numerical examples
Mutual synchronization and clustering in randomly coupled chaotic dynamical networks
We introduce and study systems of randomly coupled maps (RCM) where the
relevant parameter is the degree of connectivity in the system. Global
(almost-) synchronized states are found (equivalent to the synchronization
observed in globally coupled maps) until a certain critical threshold for the
connectivity is reached. We further show that not only the average
connectivity, but also the architecture of the couplings is responsible for the
cluster structure observed. We analyse the different phases of the system and
use various correlation measures in order to detect ordered non-synchronized
states. Finally, it is shown that the system displays a dynamical hierarchical
clustering which allows the definition of emerging graphs.Comment: 13 pages, to appear in Phys. Rev.
Role of Network Topology in the Synchronization of Power Systems
We study synchronization dynamics in networks of coupled oscillators with
bimodal distribution of natural frequencies. This setup can be interpreted as a
simple model of frequency synchronization dynamics among generators and loads
working in a power network. We derive the minimum coupling strength required to
ensure global frequency synchronization. This threshold value can be
efficiently found by solving a binary optimization problem, even for large
networks. In order to validate our procedure, we compare its results with
numerical simulations on a realistic network describing the European
interconnected high-voltage electricity system, finding a very good agreement.
Our synchronization threshold can be used to test the stability of frequency
synchronization to link removals. As the threshold value changes only in very
few cases when aplied to the European realistic network, we conclude that
network is resilient in this regard. Since the threshold calculation depends on
the local connectivity, it can also be used to identify critical network
partitions acting as synchronization bottlenecks. In our stability experiments
we observe that when a link removal triggers a change in the critical
partition, its limits tend to converge to national borders. This phenomenon,
which can have important consequences to synchronization dynamics in case of
cascading failure, signals the influence of the uncomplete topological
integration of national power grids at the European scale.Comment: The final publication is available at http://www.epj.org (see
http://www.springerlink.com/content/l22k574x25u6q61m/
- …