4,109 research outputs found
Synchronization under matrix-weighted Laplacian
Synchronization in a group of linear time-invariant systems is studied where
the coupling between each pair of systems is characterized by a different
output matrix. Simple methods are proposed to generate a (separate) linear
coupling gain for each pair of systems, which ensures that all the solutions
converge to a common trajectory. Both continuous-time and discrete-time cases
are considered.Comment: 21 page
Bounding network spectra for network design
The identification of the limiting factors in the dynamical behavior of
complex systems is an important interdisciplinary problem which often can be
traced to the spectral properties of an underlying network. By deriving a
general relation between the eigenvalues of weighted and unweighted networks,
here I show that for a wide class of networks the dynamical behavior is tightly
bounded by few network parameters. This result provides rigorous conditions for
the design of networks with predefined dynamical properties and for the
structural control of physical processes in complex systems. The results are
illustrated using synchronization phenomena as a model process.Comment: 17 pages, 4 figure
Synchronization of small oscillations
Synchronization is studied in an array of identical oscillators undergoing
small vibrations. The overall coupling is described by a pair of
matrix-weighted Laplacian matrices; one representing the dissipative, the other
the restorative connectors. A construction is proposed to combine these two
real matrices in a single complex matrix. It is shown that whether the
oscillators synchronize in the steady state or not depends on the number of
eigenvalues of this complex matrix on the imaginary axis. Certain refinements
of this condition for the special cases, where the restorative coupling is
either weak or absent, are also presented.Comment: 16 pages, 6 figure
On recovery guarantees for angular synchronization
The angular synchronization problem of estimating a set of unknown angles
from their known noisy pairwise differences arises in various applications. It
can be reformulated as a optimization problem on graphs involving the graph
Laplacian matrix. We consider a general, weighted version of this problem,
where the impact of the noise differs between different pairs of entries and
some of the differences are erased completely; this version arises for example
in ptychography. We study two common approaches for solving this problem,
namely eigenvector relaxation and semidefinite convex relaxation. Although some
recovery guarantees are available for both methods, their performance is either
unsatisfying or restricted to the unweighted graphs. We close this gap,
deriving recovery guarantees for the weighted problem that are completely
analogous to the unweighted version.Comment: 20 pages, 5 figure
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