An instability theorem for nonlinear fractional differential systems
Nguyen Dinh Cong Doan Thai Son Stefan Siegmund Hoang The Tuan
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
$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$
$α∈ (0,1)$
is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.
keywords: Fractional differential equations qualitative theory stability theory instability condition
On integral separation of bounded linear random differential equations
Nguyen Dinh Cong Doan Thai Son
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.
keywords: Lyapunov exponents multiplicative ergodic theorem Random differential equations integral separation genericity.
On the spectrum of the one-dimensional Schrödinger operator
Nguyen Dinh Cong Roberta Fabbri
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.
keywords: Quasi-periodic Schrödinger operator Cantor spectrum exponential dichotomy.
Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term
Nguyen Dinh Cong Nguyen Thi Thuy Quynh
In this paper we show that under a nondegeneracy condition Lyapunov exponents and central exponents of linear Ito stochastic di erential 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 di erential equation which is the deterministic part of the system.
keywords: central exponents Lyapunov exponents Lyapunov spectrum two-parameter stochastic ow nonautonomous stochastic di erential equation
On Lyapunov exponents of difference equations with random delay
Nguyen Dinh Cong Thai Son Doan Stefan Siegmund
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.
keywords: Random difference equations random delay multiplicative ergodic theorem Lyapunov exponent.
The factorization method for the Drude-Born-Fedorov model for periodic chiral structures
Dinh-Liem Nguyen
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.
keywords: Periodic chiral structures factorization method diffraction gratings. Drude-Born-Fedorov model inverse scattering
Imaging with electromagnetic waves in terminating waveguides
Liliana Borcea Dinh-Liem Nguyen
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.
keywords: imaging terminating waveguide Maxwell's equations. Electromagnetic inverse scattering
A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data
Michael V. Klibanov Dinh-Liem Nguyen Loc H. Nguyen Hui Liu

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.

keywords: Global convergence coefficient inverse problem inverse medium problem coefficient identification multi-frequency data experimental data single measurement

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