65 research outputs found
Pseudospectrum of horizonless compact objects: a bootstrap instability mechanism
Recent investigations of the pseudospectrum in black hole spacetimes have
shown that quasinormal mode frequencies suffer from spectral instabilities.
This phenomenon may severely affect gravitational-wave spectroscopy and limit
precision tests of general relativity. We extend the pseudospectrum analysis to
horizonless exotic compact objects which possess a reflective surface
arbitrarily close to the Schwarzschild radius, and find that their quasinormal
modes also suffer from an overall spectral instability. Even though all the
modes themselves decay monotonically, the pseudospectrum contours of equal
resonance magnitude around the fundamental mode and the lowest overtones can
cross the real axis into the unstable regime of the complex plane, unveiling
the existence of nonmodal pseudo-resonances. A pseudospectrum analysis further
predicts that fluctuations to the system may destabilize the object when next
to leading-order effects are considered, as the triggering of pseudo-resonant
growth can break the order-expansion of black-hole perturbation theory.Comment: 19 pages, 5 figures; v2: minor changes and references adde
Approximating the Real Structured Stability Radius with Frobenius Norm Bounded Perturbations
We propose a fast method to approximate the real stability radius of a linear
dynamical system with output feedback, where the perturbations are restricted
to be real valued and bounded with respect to the Frobenius norm. Our work
builds on a number of scalable algorithms that have been proposed in recent
years, ranging from methods that approximate the complex or real pseudospectral
abscissa and radius of large sparse matrices (and generalizations of these
methods for pseudospectra to spectral value sets) to algorithms for
approximating the complex stability radius (the reciprocal of the
norm). Although our algorithm is guaranteed to find only upper bounds to the
real stability radius, it seems quite effective in practice. As far as we know,
this is the first algorithm that addresses the Frobenius-norm version of this
problem. Because the cost mainly consists of computing the eigenvalue with
maximal real part for continuous-time systems (or modulus for discrete-time
systems) of a sequence of matrices, our algorithm remains very efficient for
large-scale systems provided that the system matrices are sparse
Non-Normality In Scalar Delay Differential Equations
Thesis (M.S.) University of Alaska Fairbanks, 2006Analysis of stability for delay differential equations (DDEs) is a tool in a variety of fields such as nonlinear dynamics in physics, biology, and chemistry, engineering and pure mathematics. Stability analysis is based primarily on the eigenvalues of a discretized system. Situations exist in which practical and numerical results may not match expected stability inferred from such approaches. The reasons and mechanisms for this behavior can be related to the eigenvectors associated with the eigenvalues. When the operator associated to a linear (or linearized) DDE is significantly non-normal, the stability analysis must be adapted as demonstrated here. Example DDEs are shown to have solutions which exhibit transient growth not accounted for by eigenvalues alone. Pseudospectra are computed and related to transient growth
Stable high-order finite-difference methods based on non-uniform grid point distributions
It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstr¨om theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q_N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q = N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q<N through the use of the proposed non-uniform grids. Discussions are complemented by spectra and pseudospectra of the finite-difference operators
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