Wideband chaotic microwave signal generation in a ring system with a nonlinear delay line on coupled ferromagnetic films

Various regimes of chaotic signal generation have been studied in an self-oscillating ring system with a wideband nonlinear delay line based on coupled ferromagnetic films, in which magnetostatic backward volume waves (MSBVWs) are excited. It is shown for the first time that the simultaneous excitation of two waves in the coupled films leads to the appearance of a falling part in the dynamic characteristic of the MSBVW delay line and the generation of a broadband chaotic signal with a continuous spectrum. Under certain conditions, the ring exhibits a regime of intermittency, in which broadband chaotic signals with both continuous and discrete spectra are generated.     

Influence of the amplitude and phase nonlinearity of a spin-wave delay line on the wideband chaotic microwave generation

Regimes of wideband chaotic microwave generation have been studied in a self-oscillating ring system with a nonlinear delay line, in which magnetostatic backward volume waves (MSBVWs) are excited. It is established that a change in length of the delay line influences on its amplitude and phase nonlineartity, thus making it possible to control the characteristics of the chaotic microwave signal generated in the ring. It is demonstrated for the first time that a wideband chaotic microwave signal with a continuous spectrum and a nearly Gaussian probability density distribution can be generated by exciting MSBVWs in a self-oscillating ring system with a nonlinear delay line based on single ferromagnetic film.     

Electromagnetic transverse electric-wave inverse scattering of a conductor by the genetic algorithm

The genetic algorithm is used to reconstruct the shape of a perfectly conducting cylinder illuminated by transverse electric (TE) waves. A cylinder of unknown shape scatters the incident TE wave in a free space and the scattered field is recorded outside. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the global extreme solution of the cost function. Numerical results demonstrated that the genetic algorithm can tackle the inverse problem of a larger scatterer. Even when the electrical dimension of the scatterer exceeds one wavelength and the initial guess is far from the exact one, good reconstruction was obtained. In such a case, gradient-based methods often get stuck in a local extreme. In addition, the effect of Gaussian noise on the reconstruction results is investigated. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 388–394, 1998     

Arbitrary order edge elements for electromagnetic scattering simulations using hybrid meshes and a PML

The use of arbitrary order edge elements for the simulation of two‐dimensional electromagnetic scattering problems on hybrid meshes of triangles and quadrilaterals is described. Single‐frequency incident waves, generated by a source in the far field, are considered and the solution is determined in the frequency domain. For numerical simulation, the solution domain is truncated at a finite distance from the perfectly conducting scatterer and the non‐reflecting boundary condition at the truncated boundary is imposed by the use of a perfectly matched layer (PML). Several examples are included to demonstrate the performance of the proposed procedure. Copyright © 2002 John Wiley & Sons, Ltd.     

Intensity-intensity fluctuations of stochastic fields produced upon weak scattering

We derive expressions for the intensity-intensity correlations of a field produced on scattering of a random field governed by Gaussian statistics from a particle with a deterministic or a random refractive index distribution. Our results generalize the ones in [Opt. Lett. 35, 4000, 2010] to the case of an arbitrarily correlated incident field. We consider as specific examples the cases of a single plane wave and two partially correlated plane waves incident upon the scatterer.     

Experimental investigation of a light-controlled nonlinear scatterer

It is demonstrated that the spectrum of a high-frequency signal reradiated by a nonlinear scatterer can be controlled using optical radiation. The scatterer is a monopole antenna with a nonlinear element (phototransistor). By illuminating the phototransistor, it is possible to change its nonlinear conductance and capacitance and thereby the spectrum of the scattered signal. The low-frequency (LF) modulation of parameters of the nonlinear element caused by the modulated optical radiation results in modulation of the scattered signal both at the base frequency and the combination frequencies produced by the nonlinear conversion. The proposed scatterer can be used as a feederless sensor of the electric field whose output signal can be readily separated from interferences due to the LF modulation.     

Generation of chaotic microwave pulses in broadband self-oscillating ring system with ferromagnetic film under the action of external pulse-modulated microwave signal

We have experimentally studied specific features of the generation of chaotic microwave pulse trains in a self-oscillating ring system with nonlinear delay line on surface magnetostatic waves, bandpass filter, and power amplifier on GaAs field-effect transistors under the action of an external pulse-modulated microwave signal occurring outside the band of the generated chaotic signal. It is established that a decrease in the off/duty ratio in the external pulse-modulated microwave signal leads to an increase in this ratio for the chaotic microwave pulses. The integral power of the chaotic microwave signal generated under the pulsed external signal action is increased as compared to the power of signal generated in the autonomous regime.     

横浪中船舶的随机混沌运动

采用概率密度函数和数值模拟的方法研究随机横浪中船舶的混沌运动特性和发生混沌运动的临界参数条件。综合考虑非线性阻尼、非线性恢复力矩以及白噪声横浪激励,建立了船舶的横摇非线性随机微分方程。用随机Melnikov均方准则确定混沌运动的系统参数域后,应用路径积分法求解随机微分方程得到了响应的概率密度函数。研究发现:当噪声强度大于混沌临界值时,船舶出现随机混沌运动;对于高的白噪声激励强度,系统响应有两种较大可能的状态并在这两个状态间随机跳跃,这时船舶的运动不稳定并可能发生倾覆。     

Improving the accuracy of experimental prediction of the detectability of nonlinear objects by radar

The possibility of calibration of nonlinear radar measuring systems using a physical standard scatterer is investigated. Suggestions on the shape of such a scatterer at super-high frequencies are formulated. Translated from Izmeritel'naya Tekhnika, No. 11, pp. 44–48, November, 1999.     

Theory of a fixed scatterer embedded in a turbid medium: numerical results

Based on a previous theory of diffuse photon density waves by Furutsu [J. Opt. Soc. Am. A 15, 1371 (1998)], several sets of figures are prepared to detect a fixed scatterer (object) embedded in a turbid layer, such as a tumor in tissue, with a source and a detector placed independently along the boundaries on different sides. The relative total intensity of the wave is introduced such that it is reduced to 1 in the case of no scatterer and usually less than that, owing to a shadowing by the scatterer. Sets of curves are presented to demonstrate shadow images of the scatterer observed along the layer boundaries depending on the scatterer's location.     

The linear sampling method for a mixed scattering problem

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by a mixed scatterer, which is a combination of an open crack and an impenetrable obstacle. Firstly, the well-posedness of the solution to the direct scattering problem is established using the variational method. Then, a uniqueness result for the inverse scattering problem is proved, that is, both of the crack and the impenetrable obstacle can be uniquely determined simultaneously by the knowledge of the far-field pattern. Furthermore, a mathematical basis is given to reconstruct the shape of the crack and the impenetrable obstacle using the linear sampling method, and some numerical examples are given to establish the viability of our approach.     

Scattering of a plane harmonic SH wave by a completely embedded corrugated scatterer

Anti‐plane‐strain model for steady‐state scattering of elastic waves by a rough inclusion or a cavity embedded in a half space is considered by using a direct boundary integral equation method. The roughness of the scatterer is assumed to be periodic with arbitrary amplitude and period. Detailed testing of the numerical results is presented. The motion along the half‐space surface is evaluated for different corrugations, frequencies and impedance contrast of the materials. The importance of the scatterer roughness upon the displacement field is clearly demonstrated. It was shown that larger corrugation amplitudes, shorter corrugation periods and higher frequencies may produce significant change in the displacement field when compared with the corresponding smooth scatterer result. This effect strongly depends upon the impedance contrast of the materials. Copyright © 2008 John Wiley & Sons, Ltd.     

Semi-inverse methods for obtaining partial differential equations with chaotic solutions

Semi-inverse methods are used to construct partial differential equations that have chaotic solutions. For first order P.D.E., this comes about by the natural association of the P.D.E. with the orbital equations for the characteristics. Explicit examples are given of linear, quasilinear, nonlinear and systems with the same principal part. The latter kind are used to show that there are 2-dimensional, steady, incompressible, inviscid fluid flows that exhibit chaotic behavior. Second order P.D.E. Are obtained through the requirement that they admit similarity solutions that satisfy O.D.E. With known chaotic behavior. Explicit systems of nonlinear second order P.D.E. That admit chaotic simple wave similarity solutions are constructed.     

Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a honeycomb sandwich plate

The global bifurcations and multi-pulse chaotic dynamics of a simply supported honeycomb sandwich rectangular plate under combined parametric and transverse excitations are investigated in this paper for the first time. The extended Melnikov method is generalized to investigate the multi-pulse chaotic dynamics of the non-autonomous nonlinear dynamical system. The main theoretical results and the formulas are obtained for the extended Melnikov method of the non-autonomous nonlinear dynamical system. The nonlinear governing equation of the honeycomb sandwich rectangular plate is derived by using the Hamilton’s principle and the Galerkin’s approach. A two-degree-of-freedom non-autonomous nonlinear equation of motion is obtained. It is known that the less simplification processes on the system will result in a better understanding of the behaviors of the multi-pulse chaotic dynamics for high-dimensional nonlinear systems. Therefore, the extended Melnikov method of the non-autonomous nonlinear dynamical system is directly utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the two-degree-of-freedom non-autonomous nonlinear system for the honeycomb sandwich rectangular plate. The theoretical results obtained here indicate that multi-pulse chaotic motions can occur in the honeycomb sandwich rectangular plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the honeycomb sandwich rectangular plate. It also demonstrates the validation of the theoretical prediction.     

Axisymmetric vibrations of elastic-plastic cylindrical shells by Galerkin's method

The paper discusses nonlinear axisymmetric vibrations of elastic-plastic cylindrical shells. The shell is compressed initially by an axial load and then fixed in its compressed position. After that a short-pulse or periodical lateral load is applied. The problem in question is solved by the Galerkin's method. An algorithm, which allows to get solutions for an arbitrary number of degrees of freedom, is proposed. Numerical examples are given and the possibility of chaotic vibrations is discussed.     

A meshless numerical identification of a sound-hard obstacle

We propose a simple meshless method for detecting a rigid (sound-hard) scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered near field. This inverse problem is ill-posed since a solution may not be unique and furthermore, small errors in the input data cause large errors in the output solution. We develop a nonlinear minimization regularized method of fundamental solutions (MFS) for obtaining the numerical solution of the inverse problem in question. Although the MFS is restricted to homogeneous media with constant wavenumber, it is easy to use and simple to implement in higher dimensions. The proposed scheme is tested on several numerical examples and its stability is investigated by inverting measurements contaminated by random noise.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Summary. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamiltons equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.     

Spectra and autocorrelation functions of a geometrically nonlinear beam

Abstract

The aim of this paper is to explore the Fourier spectra and autocorrelation functions of a geometrically nonlinear elastic beam with supports subjected to harmonic excitation. By using the large deformation theory and considering the deflection caused by both moment and shear as well as the inertia force induced by both displacement and rotation, the equation of motion is derived, which turns out to be a nonlinear fourth‐order partial differential equation. Because of the complexity of the partial differential equation, this paper first employs the Galerkin method to reduce it into a nonlinear second‐order ordinary differential equation. Then the Runge‐Kutta‐Verner numerical method is applied to find the solution. Due to the nonlinearity of the equation, chaotic motion is found to exist in this nonlinear elastic beam system. This paper draws time histories and Poincaré maps to show that chaotic motion indeed exists in the geometrically nonlinear elastic beam. For both regular and chaotic motions, Fourier spectra and autocorrelation functions are then plotted, which can be used as important criteria to diagnose whether the geometrically nonlinear beam will be chaotic in the future because of the significantly different results from those two kinds of motions.     

桩基非线性轴向振动的多时间尺度分析

假定桩基及桩周土的材料满足非线性弹性和线性粘弹性本构关系，给出了分析桩基轴向振动的非线性偏微分方程。用多时间尺度法推导出一端固定、另一端自由的桩基非线性轴向自由振动的n-阶主频率和位移的近似表达式，给出了数值算例，考察了参数的影响。研究结果表明：非线性系统的n-阶主频率不仅与线性系统的n-阶固有频率有关，而且也与振幅、阻尼系数和材料的非线性特征量有关；系统的响应中除含有主频率的谐波外，还含有2倍、3倍的主频率的高次谐波和二个或三个主频率的和或差的谐波存在。