14,276 research outputs found
On the stability of cycles by delayed feedback control
We present a delayed feedback control (DFC) mechanism for stabilizing cycles
of one dimensional discrete time systems. In particular, we consider a delayed
feedback control for stabilizing -cycles of a differentiable function of the form where
with
. Following an approach of Morg\"ul, we construct a map
whose fixed points
correspond to -cycles of . We then analyze the local stability of the
above DFC mechanism by evaluating the stability of the corresponding equilibrum
points of . We associate to each periodic orbit of an explicit
polynomial whose Schur stability corresponds to the stability of the DFC on
that orbit. An example indicating the efficacy of this method is provided
Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed
Stabilizing unstable periodic orbits in a chaotic invariant set not only
reveals information about its structure but also leads to various interesting
applications. For the successful application of a chaos control scheme,
convergence speed is of crucial importance. Here we present a predictive
feedback chaos control method that adapts a control parameter online to yield
optimal asymptotic convergence speed. We study the adaptive control map both
analytically and numerically and prove that it converges at least linearly to a
value determined by the spectral radius of the control map at the periodic
orbit to be stabilized. The method is easy to implement algorithmically and may
find applications for adaptive online control of biological and engineering
systems.Comment: 21 pages, 6 figure
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
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