Aperiodic behaviour of baroclinic wave packets

The amplitude evolution equation for weakly non-linear wave packets in a continuously stratified and sheared baroclinic flow, in which both dissipation and dispersion effects are strong, is shown to be the generalised time-dependent Ginzburg-Landau equation. This equation possesses both periodic and aperiodic solutions, depending critically on parameter values.     

Circumferential and helical normal waves of a cylindrical waveguide: Helical waves in a free space

Properties of circumferential and helical normal waves of a cylindrical waveguide, which appear as aperiodic (in angle) solutions to the Helmholtz equation, are considered. The dispersion characteristics and eigenfunctions of these wave are determined for the Dirichlet and Neumann boundary conditions, and the spatial structure of these waves is described in detail. The properties of helical waves in a free space with cylindrical symmetry are considered.     

New classes of exact solutions and some transformations of the Navier-Stokes equations

New classes of exact solutions of three-dimensional nonstationary Navier-Stokes equations are described. These solutions contain arbitrary functions. Many periodic solutions (both with respect to the spatial coordinate and with respect to time) and aperiodic solutions are obtained, which can be expressed in terms of elementary functions. A Crocco-type transformation is presented, which reduces the order of the equation for the longitudinal component of the velocity. Problems concerning the nonlinear stability/instability of the solutions thus obtained are investigated. It turns out that a specific feature of many solutions of the Navier-Stokes equations is their instability. It is shown that instability can take place not only for rather large Reynolds numbers but also for arbitrarily small ones (and can be independent of the velocity profile of the fluid). A general physical interpretation and classification of solutions is given.     

Dynamic features of stimulated raman atomic-molecular conversion in a mixture of two bose gases with the formation of bose condensates of heteronuclear molecules

Based on the mean-field approximation, a system of nonlinear differential equations is obtained which describes the dynamics of atomic-molecular conversion induced by two resonance Raman laser pulses. The obtained integrals of motion allow one to reduce the problem to one evolution equation for the density of molecules. The solutions of this equation show that different time evolution regimes of the system are possible, depending on the initial conditions: the periodic and aperiodic regimes and the rest. The dependences of the period and amplitude on the initial density of particles and the initial phase difference are studied in detail. The possibility of the phase control of the system dynamics is proved.     

On the transition to aperiodic motion in linear systems with strong relaxation

The Bloch equations are shown to possess purely aperiodic solutions at intermediate values of the transverse relaxation coefficient only. We present also a simple example of a mechanical system where the regions of aperiodic motion are separated by the region of damped oscillations.     

Modeling of heat explosion with convection

The work is devoted to numerical simulations of the interaction of heat explosion with natural convection. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Navier-Stokes equations under the Boussinesq approximation. We show how complex regimes appear through successive bifurcations leading from a stable stationary temperature distribution without convection to a stationary symmetric convective solution, stationary asymmetric convection, periodic in time oscillations, and finally aperiodic oscillations. A simplified model problem is suggested. It describes the main features of solutions of the complete problem.     

Optimal Design of Multilayer Mirrors for Water-Window Microscope Optics

A small bandwidth of periodic multilayers at wavelengths 2.4–4.4 nm presents problems for the spectral matching of mirrors. This leads to low throughput of a Schwarzschild microscope and its sensitivity to technological errors in layer thickness. We consider two cures for these difficulties: aperiodic coatings and a lateral gradient of layer thickness. Our design of aperiodic multilayers maximizes the throughput of a microscope made at a fixed level of technology. The method includes a new merit function, fast procedure for its minimization, linkage between multilayers and an equation for a lateral gradient of layer thickness. Computation is performed for Sc/Cr coatings at 398 eV. It shows that the aperiodicity makes optics stable to technological errors, while the lateral gradient increases throughput, but enhances sensitivity to the errors. The best results are obtained for aperiodic mirrors with a lateral gradient of thickness, which assure both high microscope throughput and stability to the errors.     

Bifurcations and loss of orbital stability in nonlinear viscoelastic beam arrays subject to parametric actuation

The collective dynamic response of microbeam arrays is governed by nonlinear effects, which have not yet been fully investigated and understood. This work employs a nonlinear continuum-based model in order to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled micro-electromechanical beams that are parametrically actuated. Investigations focus on the behavior of small size arrays in the one-to-one internal resonance regime, which is generated for low or zero DC voltages. The dynamic equations of motion of a two-element system are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Analytically obtained results are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic responses of the two- and three-beam systems reveal coexisting periodic and aperiodic solutions. The stability analysis enables construction of a detailed bifurcation structure, which reveals coexisting stable periodic and aperiodic solutions. For zero DC voltage only quasi-periodic and no evidence for the existence of chaotic solutions are observed. This study of small size microbeam arrays yields design criteria, complements the understanding of nonlinear nearest-neighbor interactions, and sheds light on the fundamental understanding of the collective behavior of finite-size arrays.     

A study of laser rate equations by Liapunov's second method

An investigation is made of a system of coupled nonlinear differential equations (Statz-DeMars equations), describing the time variation of photon density and inversion in a laser or maser, without solving these equations explicitly. The method applied is based onLiapunov's stability theory. The results are rigorous and imply no approximation; they are, therefore, valid for arbitrarily large nonlinear terms. In the physically meaningful halfspace of the phase plane, i.e. where the photon density is not negative, the Statz-DeMars equations admit only damped periodic and damped aperiodic solutions. The transition between the aperiodic and the periodic mode is achieved, when the pumping rate exceeds a critical value. It is proven that the whole halfspace considered belongs to the domain of asymptotic stability of the equilibrium state and, therefore, no limit cycles and no diverging solutions exist.     

Helical normal waves near a cylindrical cavity in an elastic medium

The properties of helical waves arising near a cylindrical cavity in an elastic medium are described. These waves are a manifestation of aperiodic (in angle) solutions to the dynamic elasticity equations. The dispersion characteristics of the waves are determined, and the spatial structure of the waves is described.     

Deterministic mode alternation,giant pulses and chaos in a bidirectional CO2 ring laser

A CO₂ ring laser with a single longitudinal mode propagating in each direction shows a variety of stable, periodic, and aperiodic phenomena depending on gas pressure, cavity detuning and relative excitation. Three distinct low frequency time scales for dynamical behavior are observed and are explained by numerical solutions of an appropriate model.     

Absolute parametric instability in a nonuniform plane plasma waveguide

The paper reports an analysis of the effect of spatial plasma nonuniformity on absolute parametric instability (API) of electrostatic waves in magnetized plane waveguides subjected to an intense high-frequency (HF) electric field using the separation method. In this case the effect of strong static magnetic field is considered. The problem of strong magnetic field is solved in 1D nonuniform plane plasma waveguide. The equation describing the spatial part of the electric potential is obtained. Also, the growth rates and conditions of the parametric instability for periodic and aperiodic cases are obtained. It is found that the spatial nonuniformity of the plasma exerts a stabilizing effect on the API. It is shown that the growth rates of periodic and aperiodic API in nonuniform plasma are less compared to that of uniform plasma.     

Phenomenological theory of damped and undamped intensity oscillations in lasers

The usual phenomenological laser rate equations for the population inversion and the light intensity (Statz-deMars equations) are generalized to take into account intensity-dependent losses and Einstein coefficients. Sufficient conditions are given for these generalized equations in order to have only physically admissible solutions. The solutions either tend towards an equilibrium value in an aperiodic or oscillatory way or display an undamped periodic behavior. The theory is applied to the three-level laser and the four-level laser, without and with a saturable absorber asQ-switch. The applicability of the theory to various problems is discussed.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

一维六方准晶非周期平面内的平面应变理论

针对一维六方准晶中一种新平面弹性问题, 即非周期平面内的平面弹性问题, 通过引入应力势函数, 建立了一维六方准晶中非周期平面内的平面应变理论. 作为应用, 求解了一维六方准晶中垂直于准周期方向的椭圆孔口问题, 得到了其弹性应力场的解析解. 在极限情形下, 可给出裂纹问题的解. 关键词： 一维六方准晶 椭圆孔口 广义解析函数     

Model of electron beam transformation in a narrow tube

The problem of transformation of an electron beam in a long narrow conducting tube is considered. General wave solutions for the propagation of an arbitrary (including aperiodic) modulation of a homogeneous beam are obtained in the case of weak modulation. The solution corresponding to harmonic modulation is compared to the solutions obtained earlier and described in the literature. The problem of beam transformation corresponding to the bunching regime in high-power klystrons is also considered (including overtaking). A discrete analytic solution to the problem is obtained in the ??frozen beam approximation.?? This solution is tested for the problem of spilling of a monovelocity bunch and for the problem of the efficiency of a two-cavity klystron with infinitely narrow gaps. In both cases, physically correct results are obtained.     

Extended spin waves in aperiodic ferromagnetic chains

In this work, a one-dimensional quantum Heisenberg ferromagnet with aperiodic exchange couplings is considered. To produce an aperiodic distribution of exchange couplings, it is used a sinusoidal function whose phase φ varies as a power-law, φ∝n^ν, where n labels the positions along the chain. By using exact diagonalization, the spin-wave participation number and the local density of states are computed. The numerical calculations indicate that for 0 < ν < 1, this ferromagnetic system displays a phase of extended spin waves in the low-energy region. For ν > 1 all spin waves are localized except for the zero energy mode. By integrating the time-dependent Schr?dinger equation, the temporal evolution of the mean-square displacement of the wave-packet was followed. Associated with the emergence of extended spin waves, it was observed that the wave-packet mean-square displacement displays a ballistic spread.     

Bifurcations in a model of magnetoconvection

Bifurcations from oscillatory solutions are studied in a truncated model of two-dimensional Boussinesq magnetoconvection. The fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period-doubling cascade. After the accumulation point there is a semiperiodic cascade leading to chaotic behaviour. Then the semiperiodic cascade is repeated in reverse, followed by a period-halving cascade and a bifurcation back to symmetry. Finally, the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle-foci. The interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions, each with its own cascade. This pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle-foci with eigenvalues that satisfy Shil'nikov's inequality.     

体全息成像系统中非周期体光栅的深度选择性研究

由一束球面波和一束平面波或者两束球面波干涉所形成的体全息光栅,由于其光栅矢量在全息图体积内是随位置的变化而变化的,所以被称为非周期型的体光栅。主要研究了用于体全息三维成像系统的非周期体光栅的深度选择特性。根据叠加的原理,将非周期体光栅看作多个固定周期的基元体光栅的叠加,结合耦合波理论分析非周期体光栅的衍射特性。采用这一方法,利用Matlab软件模拟,研究了两束记录光夹角对所记录的非周期体光栅深度选择性的影响和在两束记录光夹角相同时,球面参考光体全息成像系统及平面参考光体全息成像系统深度选择性的差别,最后在光折变晶体材料中进行非周期型和周期型体光栅的记录和再现,对模拟结果进行了实验验证。     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.