400 research outputs found
Sensing and Control in Symmetric Networks
In engineering applications, one of the major challenges today is to develop
reliable and robust control algorithms for complex networked systems.
Controllability and observability of such systems play a crucial role in the
design process. The underlying network structure may contain symmetries --
caused for example by the coupling of identical building blocks -- and these
symmetries lead to repeated eigenvalues in a generic way. This complicates the
design of controllers since repeated eigenvalues might decrease the
controllability of the system. In this paper, we will analyze the relationship
between the controllability and observability of complex networked systems and
graph symmetries using results from representation theory. Furthermore, we will
propose an algorithm to compute sparse input and output matrices based on
projections onto the isotypic components. We will illustrate our results with
the aid of two guiding examples, a network with symmetry and the
Petersen graph
Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
Numerical continuation methods for deterministic dynamical systems have been
one of the most successful tools in applied dynamical systems theory.
Continuation techniques have been employed in all branches of the natural
sciences as well as in engineering to analyze ordinary, partial and delay
differential equations. Here we show that the deterministic continuation
algorithm for equilibrium points can be extended to track information about
metastable equilibrium points of stochastic differential equations (SDEs). We
stress that we do not develop a new technical tool but that we combine results
and methods from probability theory, dynamical systems, numerical analysis,
optimization and control theory into an algorithm that augments classical
equilibrium continuation methods. In particular, we use ellipsoids defining
regions of high concentration of sample paths. It is shown that these
ellipsoids and the distances between them can be efficiently calculated using
iterative methods that take advantage of the numerical continuation framework.
We apply our method to a bistable neural competition model and a classical
predator-prey system. Furthermore, we show how global assumptions on the flow
can be incorporated - if they are available - by relating numerical
continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size
restrictions]; v2 - added Section 9 on Kramers' formula, additional
computations, corrected typos, improved explanation
Generic bifurcation of Hamiltonian vector fields with symmetry
One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. We classify this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart) or definiteness properties of the Hamiltonian (Cartan-Oh) to determine whether the eigenvalues split or pass on the imaginary axis
Observing the Symmetry of Attractors
We show how the symmetry of attractors of equivariant dynamical systems can
be observed by equivariant projections of the phase space. Equivariant
projections have long been used, but they can give misleading results if used
improperly and have been considered untrustworthy. We find conditions under
which an equivariant projection generically shows the correct symmetry of the
attractor.Comment: 28 page LaTeX document with 9 ps figures included. Supplementary
color figures available at http://odin.math.nau.edu/~jws
Relatively Coherent Sets as a Hierarchical Partition Method
Finite time coherent sets [8] have recently been defined by a measure based
objective function describing the degree that sets hold together, along with a
Frobenius-Perron transfer operator method to produce optimally coherent sets.
Here we present an extension to generalize the concept to hierarchially defined
relatively coherent sets based on adjusting the finite time coherent sets to
use relative mesure restricted to sets which are developed iteratively and
hierarchically in a tree of partitions. Several examples help clarify the
meaning and expectation of the techniques, as they are the nonautonomous double
gyre, the standard map, an idealized stratospheric flow, and empirical data
from the Mexico Gulf during the 2010 oil spill. Also for sake of analysis of
computational complexity, we include an appendic concerning the computational
complexity of developing the Ulam-Galerkin matrix extimates of the
Frobenius-Perron operator centrally used here
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