8,413 research outputs found
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
Cluster synchronization in networks of coupled non-identical dynamical systems
In this paper, we study cluster synchronization in networks of coupled
non-identical dynamical systems. The vertices in the same cluster have the same
dynamics of uncoupled node system but the uncoupled node systems in different
clusters are different. We present conditions guaranteeing cluster
synchronization and investigate the relation between cluster synchronization
and the unweighted graph topology. We indicate that two condition play key
roles for cluster synchronization: the common inter-cluster coupling condition
and the intra-cluster communication. From the latter one, we interpret the two
well-known cluster synchronization schemes: self-organization and driving, by
whether the edges of communication paths lie at inter or intra-cluster. By this
way, we classify clusters according to whether the set of edges inter- or
intra-cluster edges are removable if wanting to keep the communication between
pairs of vertices in the same cluster. Also, we propose adaptive feedback
algorithms on the weights of the underlying graph, which can synchronize any
bi-directed networks satisfying the two conditions above. We also give several
numerical examples to illustrate the theoretical results
Cluster and group synchronization in delay-coupled networks
We investigate the stability of synchronized states in delay-coupled networks
where synchronization takes place in groups of different local dynamics or in
cluster states in networks with identical local dynamics. Using a master
stability approach, we find that the master stability function shows a discrete
rotational symmetry depending on the number of groups. The coupling matrices
that permit solutions on group or cluster synchronization manifolds show a very
similar symmetry in their eigenvalue spectrum, which helps to simplify the
evaluation of the master stability function. Our theory allows for the
characterization of stability of different patterns of synchronized dynamics in
networks with multiple delay times, multiple coupling functions, but also with
multiple kinds of local dynamics in the networks' nodes. We illustrate our
results by calculating stability in the example of delay-coupled semiconductor
lasers and in a model for neuronal spiking dynamics.Comment: 11 pages, 7 figure
- …