1,421 research outputs found
Unifying Dynamical and Structural Stability of Equilibriums
We exhibit a fundamental relationship between measures of dynamical and
structural stability of equilibriums, arising from real dynamical systems. We
show that dynamical stability, quantified via systems local response to
external perturbations, coincides with the minimal internal perturbation able
to destabilize the equilibrium. First, by reformulating a result of control
theory, we explain that harmonic external perturbations reflect the spectral
sensitivity of the Jacobian matrix at the equilibrium, with respect to constant
changes of its coefficients. However, for this equivalence to hold, imaginary
changes of the Jacobian's coefficients have to be allowed. The connection with
dynamical stability is thus lost for real dynamical systems. We show that this
issue can be avoided, thus recovering the fundamental link between dynamical
and structural stability, by considering stochastic noise as external and
internal perturbations. More precisely, we demonstrate that a system's local
response to white-noise perturbations directly reflects the intensity of
internal white noise that it can accommodate before asymptotic mean-square
stability of the equilibrium is lost.Comment: 13 pages, 2 figure
Lorenz-Mie theory for 2D scattering and resonance calculations
This PhD tutorial is concerned with a description of the two-dimensional
generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method
used to compute the interaction of light with arrays of cylindrical scatterers.
This theory is based on the method of separation of variables and the
application of an addition theorem for cylindrical functions. The purpose of
this tutorial is to assemble the practical tools necessary to implement the
2D-GLMT method for the computation of scattering by passive scatterers or of
resonances in optically active media. The first part contains a derivation of
the vector and scalar Helmholtz equations for 2D geometries, starting from
Maxwell's equations. Optically active media are included in 2D-GLMT using a
recent stationary formulation of the Maxwell-Bloch equations called
steady-state ab initio laser theory (SALT), which introduces new classes of
solutions useful for resonance computations. Following these preliminaries, a
detailed description of 2D-GLMT is presented. The emphasis is placed on the
derivation of beam-shape coefficients for scattering computations, as well as
the computation of resonant modes using a combination of 2D-GLMT and SALT. The
final section contains several numerical examples illustrating the full
potential of 2D-GLMT for scattering and resonance computations. These examples,
drawn from the literature, include the design of integrated polarization
filters and the computation of optical modes of photonic crystal cavities and
random lasers.Comment: This is an author-created, un-copyedited version of an article
published in Journal of Optics. IOP Publishing Ltd is not responsible for any
errors or omissions in this version of the manuscript or any version derived
from i
Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian
and are perturbations of linear dispersive equations. The unperturbed dynamical
system has a bound state, a spatially localized and time periodic solution. We
show that, for generic nonlinear Hamiltonian perturbations, all small amplitude
solutions decay to zero as time tends to infinity at an anomalously slow rate.
In particular, spatially localized and time-periodic solutions of the linear
problem are destroyed by generic nonlinear Hamiltonian perturbations via slow
radiation of energy to infinity. These solutions can therefore be thought of as
metastable states.
The main mechanism is a nonlinear resonant interaction of bound states
(eigenfunctions) and radiation (continuous spectral modes), leading to energy
transfer from the discrete to continuum modes.
This is in contrast to the KAM theory in which appropriate nonresonance
conditions imply the persistence of invariant tori. A hypothesis ensuring that
such a resonance takes place is a nonlinear analogue of the Fermi golden rule,
arising in the theory of resonances in quantum mechanics. The techniques used
involve: (i) a time-dependent method developed by the authors for the treatment
of the quantum resonance problem and perturbations of embedded eigenvalues,
(ii) a generalization of the Hamiltonian normal form appropriate for infinite
dimensional dispersive systems and (iii) ideas from scattering theory. The
arguments are quite general and we expect them to apply to a large class of
systems which can be viewed as the interaction of finite dimensional and
infinite dimensional dispersive dynamical systems, or as a system of particles
coupled to a field.Comment: To appear in Inventiones Mathematica
Linear non-normal energy amplification of harmonic and stochastic forcing in turbulent channel flow
The linear response to stochastic and optimal harmonic forcing of small coherent perturbations to the turbulent channel mean flow is computed for Reynolds numbers ranging from Re_tau=500 to Re_tau=20000. Even though the turbulent mean flow is linearly stable, it is nevertheless able to sustain large amplifications by the forcing. The most amplified structures consist of streamwise elongated streaks that are optimally forced by streamwise elongated vortices. For streamwise elongated structures, the mean energy amplification of the stochastic forcing is found to be, to a first approximation, inversely proportional to the forced spanwise wavenumber while it is inversely proportional to its square for optimal harmonic forcing in an intermediate spanwise wavenumber range. This scaling can be explicitly derived from the linearised equations under the assumptions of geometric similarity of the coherent perturbations and of logarithmic base flow. Deviations from this approximate power-law regime are apparent in the premultiplied energy amplification curves that reveal a strong influence of two different peaks. The dominant peak scales in outer units with the most amplified spanwise wavelength of while the secondary peak scales in wall units with the most amplified . The associated optimal perturbations are almost independent of the Reynolds number when respectively scaled in outer and inner units. In the intermediate wavenumber range the optimal perturbations are approximatively geometrically similar. Furthermore, the shape of the optimal perturbations issued from the initial value, the harmonic forcing and the stochastic forcing analyses are almost indistinguishable. The optimal streaks corresponding to the large-scale peak strongly penetrate into the inner layer, where their amplitude is proportional to the mean-flow profile. At the wavenumbers corresponding to the large-scale peak, the optimal amplifications of harmonic forcing are at least two orders of magnitude larger than the amplifications of the variance of stochastic forcing and both increase with the Reynolds number. This confirms the potential of the artificial forcing of optimal large-scale streaks for the flow control of wall-bounded turbulent flows
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