DCDS-B

In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector

where

is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.

DCDS-S

Our aim in this paper is to investigate the openness and denseness for the set of integrally separated systems in the space of bounded linear random differential equations equipped with the $L^{\infty}$-metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation flow of the unit circle, in which the set
of integrally separated systems is open but not dense.

DCDS-B

The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.

PROC

In this paper we show that under a nondegeneracy condition Lyapunov exponents and central exponents of linear Ito stochastic dierential equation coincide. Furthermore, as the stochastic term is small and tends to zero
the highest Lyapunov exponent tends to the highest central exponent of the
ordinary dierential equation which is the deterministic part of the system.

DCDS-B

The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.

IPI

We consider the electromagnetic inverse scattering problem for the
Drude-Born-Fedorov model for periodic chiral structures known as chiral gratings
both in $\mathbb{R}^2$ and $\mathbb{R}^3$. The Factorization method
is studied as an analytical as well as a numerical tool for solving this inverse problem.
The method constructs a simple criterion for characterizing shape of the periodic scatterer
which leads to a fast imaging algorithm. This criterion is necessary and sufficient which
gives a uniqueness result in shape reconstruction of the scatterer.
The required data consists of certain components of Rayleigh sequences of
(measured) scattered fields caused by plane incident electromagnetic waves.
We propose in this electromagnetic plane wave setting a rigorous analysis for the Factorization method.
Numerical examples in two and three dimensions are also presented for showing the efficiency of the method.

IPI

We study an inverse scattering problem for Maxwell's equations in
terminating waveguides, where localized reflectors are to be imaged
using a remote array of sensors. The array probes the waveguide with
waves and measures the scattered returns. The mathematical
formulation of the inverse scattering problem is based on the
electromagnetic Lippmann-Schwinger integral equation and an explicit
calculation of the Green tensor. The image formation is carried
with reverse time migration and with $\ell_1$ optimization.

IPI

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any *a priori* knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.