Application of the mutual-interaction method to a class of two-scatterer systems: I. Two discrete scatterers

In a recent paper we developed a formalism that fully accommodates the mutual interactions among scatterers separable by parallel planes. The total fields propagating away from these planes are the unknowns of a system of difference equations. Each scatterer is characterized by a scattering function that expresses the scattered wave amplitude as a function of the incident and scattered wavevectors for a unit-amplitude plane wave scattered from the object in isolation. This function can be derived completely from the scattered far field with the help of analytic continuation. For a two-scatterer system the mutual-interaction equations reduce to a single Fredholm integral equation of the second kind. It turns out that analytic solutions are tractable for those scattering functions that are Dirac deltas or a sum of products of separable functions of the incident and scattered wavevectors. Scattering functions for planes and isotropic scatterers, as well as electric and magnetic dipoles all possess this property and are considered. The exact scattering functions agree with results obtained by analytic continuation. This paper consists of two parts. Part I derives analytic solutions for two discrete scatterers (isotropic scatterers. electric dipoles, magnetic dipoles). Part II is devoted to scattering from an object (isotropic or dipole scatterer) near an interface separating two semi-infinite uniforn-media. Because the results in this paper are exact, the effects of near-field interactions can be assessed. The forms of the scattering solutions can be adapted to objects that are both radiating and scattering.     

Application of the mutual-interaction method to a class of two-scatterer systems: II. Scatterer near an interface

This paper applies the methodology developed in Part I to the problem of a separable scatterer near a dielectric (penetrable) or perfectly conducting (impenetrable) interface. For a penetrable interface, the scatterer may be on either side of the interface (exposed or embedded). As in Part I, the scatterer may also be an active element. Thus, our solutions extend the classic treatments of dipoles radiating near a planar dielectric interface. The mutual-interaction method accommodates a uniform half-space as an equivalent scattering plate of zero thickness that preserves amplitudes and phases of the transmitted and reflected waves. Because this scattering function necessarily includes a Dirac delta function, exact analytic solutions are possible for the class of separable scatterers, which include isotropic scatterers and electric or magnetic dipoles. The results can be interpreted within the context of image theory. Integrals similar to those derived by Sommerfeld must be evaluated to calculate the spatial fields for dielectric media; however, for highly conducting media good approximations are readily obtained.     

Application of the mutual-interaction method to a class of two-scatterer systems: II. Scatterer near an interface

Abstract

This paper applies the methodology developed in Part I to the problem of a separable scatterer near a dielectric (penetrable) or perfectly conducting (impenetrable) interface. For a penetrable interface, the scatterer may be on either side of the interface (exposed or embedded). As in Part I, the scatterer may also be an active element. Thus, our solutions extend the classic treatments of dipoles radiating near a planar dielectric interface. The mutual-interaction method accommodates a uniform half-space as an equivalent scattering plate of zero thickness that preserves amplitudes and phases of the transmitted and reflected waves. Because this scattering function necessarily includes a Dirac delta function, exact analytic solutions are possible for the class of separable scatterers, which include isotropic scatterers and electric or magnetic dipoles. The results can be interpreted within the context of image theory. Integrals similar to those derived by Sommerfeld must be evaluated to calculate the spatial fields for dielectric media; however, for highly conducting media good approximations are readily obtained.     

Coherent Transmission and Reflection by a monolayer of discrete scatterers

Coherent transmission and reflection of a plane wave through a monolayer of discrete particles are considered on the basis of simple and physically transparent formulae for the single scattering approximation (SSA) corrected by introducing a multiple scattering permittivity factor. This factor allows for multiple scattering of Waves between monolayer particles, opposite to the SSA. The multiple scattering permittivity factor is considered on the basis of the quasi-crystalline approximation (QCA) via ? matrix formalism. The multiple scattering permittivity factor and parameters for obtaining coherent transmission and reflection coefficients (the effective extinction coefficient and the transmission and reflection coefficients due to rescattering) are calculated within the scope of QCA and plotted for comparison with SSA results. The expressions for these values are simplified for small Rayleigh particles to simple analytical formulae.     

Analysis of discrete automatic control systems with varying parameters using the method of orthogonal expansions. I

An analysis of discrete automatic control systems with varying parameters is considered when the coefficients of the difference equation of the system satisfy specific conditions. The analysis is carried out by expanding the output signal in a specially formulated orthonormalized system of functions. The computation algorithm may be realized on a digital computer.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 427–437, March, 1972.     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.     

A bifurcation theory for a class of discrete time Markovian stochastic systems

We present a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this ‘dependence ratio’ is a geometric invariant of the system. By introducing an equivalence relation defined on these dependence ratios, we arrive at a bifurcation theory for which in the compact case, the set of stable, i.e. non-bifurcating, systems is open and dense. The theory is illustrated with some simple examples.     

Application of the pattern equation method in spheroidal coordinates to solving diffraction problems with highly prolate scatterers

Acoustical Physics - The equations of the diagram equation method are formulated in terms of prolate spheroidal coordinates. Examples of the convergence and stability of numerical algorithms based...     

Decentralized state-feedback chaotification method of discrete Takagi-Sugeno fuzzy systems

A new chaotification method is proposed for making an arbitrarily given discrete Takagi-Sugeno (TS) fuzzy system chaotic. Based on a given discrete TS fuzzy system, the new chaotification method uses the decentralized state-feedback control and the continuous sawtooth function, instead of the modulo operation, to construct a chaotic nonlinear system,which can generate discrete chaos with the arbitrarily desired amplitude bound. We apply the improved Marotto theorem to mathematically prove that the controlled system is chaotic in the sense of Li and Yorke. In particular, an explicit formula for the computation of chaotification parameters is obtained. A numerical example is used to illustrate the theoretical results.     

Multiple scattering of acoustic waves by a half-space of distributed discrete scatterers with modified T-matrix approach

In studying the multiple scattering of acoustic waves by a half-space of distributed discrete scatterers, the quasicrystalline approximation(QCA) approach together with the hole correction (HC) or the pair distribution functions (PDF) have been used extensively, in which a system of simultaneous equations must be solved to determine the effective propagation constant and the expansion coefficients of the coherent exciting field. In this paper, we analyse the same problem under Foldy's approximation (EFA) by using the so-called modified T-matrix approach (MTMA) which was first proposed by Twersky. Two equations in a considerably simple and clear form are obtained for determining the effective propagation constant and the amplitude of the coherent transmitted field, as the scatterers are identical spheres. The numerical results in the low-frequency limit are also discussed in brief.     

On the synchronization of a class of chaotic systems based on backstepping method

This Letter focuses on the synchronization problem of a class of chaotic systems. A synchronization method is presented based on Lyapunov method and backstepping method. Finally some typical numerical examples are given to show the effectiveness of the theoretical results.     

DYNGA: a general purpose QM-MM-MD program. I. Application to water

We present a general purpose QM-MM-MD engine (DYNGA) designed to test alternative hybrid Hamiltonians geared towards the treatment of problems of interest in structural biology including the use of experimental data constraints. In this first presentation we use DYNGA to explore the behaviour of a traditional QM-MM approach in the treatment of the water—water interaction. We find the potential energy hypersurface for the water dimer computed with the HF 4–31G*/TIP3P hybrid Hamiltonian tends to be too flat. We also explore the effect of using traditional QM-MM techniques on proton wires and conclude there is a need for improvement, possibly addressed by using polarizable force fields.     

Intensity fluctuations of light propagating in system of large scatterers. I. Single scattering

An autocorrelation function for wave-field fluctuations detected by a square-law receiver behind a layer of large scatterers is found by means of nonclassical photometry in a single-scattering approximation.     

Scattering of electromagnetic waves from a half-space of randomly distributed discrete scatterers and polarized backscattering ratio law

Summary The effective-medium approximation is applied to investigate scattering from a half-space of randomly and densely distributed discrete scatterers. Starting from vector wave equations, an approximation, called effective-medium Born approximation, a particular way, treating Green's functions, and special coordinates, of which the origin is set at the field point, are used to calculate the bistatic- and back-scatterings. An analytic solution of backscattering with closed form is obtained and it shows a depolarization effect. The theoretical results are in good agreement with the experimental measurements in the cases of snow, multi- and first-year sea-ice. The root product ratio of polarization to depolarization in backscattering is equal to 8; this result constitutes a law about polarized scattering phenomena in the nature.