Thermodynamic stability of a weakly interactingFermi gas trapped in a harmonic potential

Based on the theoretical results derived from pseudopotential method and local approximation, this paper studies the thermodynamic stability of a weakly interacting Fermi gas trapped in a harmonic potential by using analytical method of thermodynamics. The effects of the interparticle interactions as well as external potential on the thermodynamic stability of the system are discussed. It is shown that the system is stable as for the complete average, but as for local parts, the system is unstable anywhere. This instability shows that the stability conditions of mechanics cannot be satisfied anywhere, and the stability conditions of thermostatics cannot be satisfied somewhere. In addition, the interactions and external potential have direct effects on the local stability of the system.     

Stability conditions of a deformed lattice

Thermodynamic stability conditions of a deformed lattice, transforming at absolute zero temperature to dynamic-type stability, are formulated. The independence of the stability condition relative to uniform deformation and dynamic stability conditions is shown. Stability conditions are derived which are violated during lattice deformation by the former.     

Stability of the ground state of a harmonic oscillator in a monochromatic wave

The stability of the ground state of a harmonic oscillator in a monochromatic wave is studied. This model describes, in particular, the dynamics of a cold ion in a linear ion trap, interacting with two laser fields with close frequencies. The stability of the "classical ground state"-the vicinity of the point (x=0,p=0)-is analyzed analytically and numerically. For the quantum case, a method for studying a stability of the quantum ground state is developed, based on the quasienergy representation. It is demonstrated that stability of the ground state may be substantially improved by increasing the resonance number, l, where l=Omega/omega+delta, Omega and omega are, respectively, the wave frequency and the oscillator frequency, l=1,2, em leader, mid R:deltamid R:<1; or by detuning the system from exact resonance, so that delta not equal 0. The influence of a large-amplitude wave (in the presence of chaos) on the stability of the ground state is analyzed for different parameters of the model in both the quantum and classical cases. (c) 2001 American Institute of Physics.     

Stability of weakly nonparaxial spatial optical solitons in a medium with a Kerr nonlinearity

A variant of perturbation theory is developed to determine the characteristics and stability of transversely two-dimensional spatial solitons in a Kerr medium under conditions of small deviations from paraxial. Distributions of the transverse and longitudinal components of the soliton electric and magnetic fields are obtained. It is shown that the power of a nonparaxial soliton in a Kerr medium increases as the propagation constant increases. A linear analysis is made of soliton stability. In addition to confirming stability, this analysis revealed “internal modes” of nonparaxial solitons and their characteristics were determined.     

Dynamic stability of a sandwich plate with a constraining layer and electrorheological fluid core

The dynamic stability problems of a sandwich plate with a constraining layer and an electrorheological (ER) fluid core subjected to an axial dynamic force are investigated. The rectangular plate is covered in an ER fluid core and a constraining layer to improve the stability of the system. Effects of the natural frequencies, static buckling loads, and loss factors on the dynamic stability behavior of the sandwich plate are studied in the paper. Rheological property of an ER material, such as viscosity, plasticity, and elasticity may be changed when applying an electric field. The modal damper and the natural frequencies for the sandwich plate are calculated for various electric fields. When an electric field is applied, the damping of the system is more effective. In this study, finite element method and the harmonic balance method are used to calculate the instability regions of the sandwich plate. The ER fluid core is found to have a significant effect on the dynamic stability regions.     

The stability of a non-extensive relativistic Fermi system

Based on the statistical theory of non-extensive relativity,and using theoretical analysis and numerical simulation,the non-extensive mechanical stability of ultra-relativistic free Fermi gas is investigated.The expressions of the stability conditions under high and low temperatures are given,and the mechanisms of the influences of temperature,ultrarelativistic effect,and non-extensive parameter q on stability are analysed.Our results show that at high temperature and under the condition of q < 1,the stability of a non-extensive system is weaker than that of an extensive system,and the relativistic effect reduces system stability as compared with a non-relativistic system.However,under the condition of q > 1,the stability of the non-extensive system is stronger than that of the extensive system,and the relativistic effect strengthens the system stability as compared with the non-relativistic system.In addition,under the condition of low temperature,the variation of the stability of the non-extensive system with temperature has a turning point.     

Relativity stability of quantum gas in a weak magnetic field

Based on the analytical expression of relativistic free energy for a weakly interacting Fermi gas in a weak magnetic field, by using the method of quantum statistics, the stability conditions of the system at both high and low temperatures are given, and the effects of magnetic field and interparticle interactions on the stability of the system are analysed. It is shown that at high temperatures, the stability conditions of the system are completely the same, no matter whether it is the ultrarelativistic case or nonrelativistic case. At extremely low temperatures, the mechanical stability conditions of the system show a similar rule through a comparison between the ultrarelativistic case and nonrelativistic case. At the same time, thermal stability of a relativistic Bose gas in a weak magnetic field is discussed, and the influence of the effect of relativity on the thermal stability of the system is investigated.     

弹性细杆平衡的动态稳定性

从动力学观点讨论弹性细杆的平衡稳定性问题.建立了弹性杆的动力学方程，导出了杆的弯扭度与截面角速度之间的运动学关系式.对具有弧坐标s和时间t双重自变量的离散动力系统扩充了Lyapunov 稳定性定义.以具有初扭率的非圆截面直杆的平衡稳定性为例，应用一次近似方法证明了当静力学意义下的稳定性条件得到满足时，动力学意义下的稳定性条件必同时满足. 关键词： 弹性杆动力学方程 Kirchhoff理论 Lyapunov稳定性     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

强磁场中Fermi气体的稳定性及顺磁性

运用理论解析和数值模拟的方法,研究强磁场中Fermi气体在低温下的稳定性及顺磁性, 分析温度、磁场对稳定性及磁化率的影响.研究结果显示,强磁场引起系统稳定性的改变, 这种改变与温度有关,而温度对稳定性改变及磁化率的影响又与自由气体的化学势即粒子数密度有关. 当自由系统的化学势为偶数时,磁场弱化系统的稳定性,温度升高可降低系统的磁化率; 当自由系统的化学势为奇数时,磁场强化系统的稳定性,温度升高可增大系统的磁化率. 升高温度,可以弱化磁场对系统稳定性的影响;加强磁场,可以使得系统稳定性的改变发生振荡, 使系统磁化率的振荡中心下移.     

Stability of gravity with a cosmological constant

The stability properties of Einstein theory with a cosmological constant Λ are investigated. For Λ > 0, stability is established for small fluctuations, about the de Sitter background, occurring inside the event horizon and semiclassical stability is analyzed. For Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics. The analysis is based on the general construction of conserved flux-integral expressions associated with the symmetries of a chosen background. The effects of an event horizon, which lead to Hawking radiation, are expressedfor general field hamiltonians. Stability for Λ < 0 is proved, using supergravity techniques, in terms of the graded anti-de Sitter algebra with spinorial charges also expressed as flux integrals.     

Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment

The effect of misalignment on the stability of two rotors connected by a flexible mechanical coupling subjected to angular misalignment is examined. The study performed is to understand the effect of angular misalignment on the stability of rotating machinery. The dimensionless stability criteria of the non-linear system of differential equations of two misaligned rigid rotors are derived using Liapunov's direct method. A rigid disk is attached at the middle of each rotor, where the rotor-disk assembly is mounted on two hydrodynamic bearings with four stiffness and four damping coefficients. Sets of dimensionless conditions for sufficient whirl stability of the two misaligned rotors are derived. The stability conditions are presented in graphical form for deeper understanding of the effect of the flexible mechanical coupling stiffness and angular misalignment on rotating machinery stability. The results show that an increase in angular misalignment or mechanical coupling stiffness terms leads to an increase of the model stability region.     

Exact region of stability for a linear orthogonal cutting model

In this work, stability of chatter in orthogonal cutting is investigated in order to identify the critical spindle speeds. The process is modeled by a second-order linear delay-differential equation and the characteristic equation is analyzed in formulating the continuous stability boundaries, explicitly for any given range of the spindle speeds. Moreover, a simple algorithm is introduced for checking whether the system is in the stable zone without determining the whole stability diagram. Numerical simulations and comparison with some other methods are presented to justify the theoretical results.     

拉曼型自由电子激光器中相对论电子运动稳定性的比较研究

拉曼型自由电子激光器作为一种兆瓦级高功率毫米波、太赫兹波辐射源, 其电子的运动稳定性对整体器件的性能至关重要.本文采用科尔莫戈罗夫熵方法, 以典型的麻省理工学院公布的实验数据为例, 比较研究拉曼型正向导引磁场和反向导引磁场两类自由电子激光器中相对论电子的运动稳定性. 结果表明:摇摆器绝热压缩磁场对电子运动的稳定性无实质性影响, 但对电子运动影响大; 电子束自身场在拉曼型正向导引磁场自由电子激光器中使电子运动稳定性变差, 而在拉曼型反向导引磁场自由电子激光器中则可改善电子运动稳定性. 关键词： 拉曼型自由电子激光器 相对论电子运动稳定性 科尔莫戈罗夫熵 电子束自身场     

Stability of a diode laser with phase-conjugate feedback

An exact analysis is presented of the steady-state stability of a semiconductor laser subjected to feedback from a phase-conjugate mirror. Reduced stability occurs at low feedback whenever the effective external delay time is an integer multiple of the relaxation oscillation period. The role of a finite response time of the mirror is to enhance drastically the steady-state stability.     

Hopf bifurcation control of a Pan-like chaotic system

This paper is concerned with the Hopf bifurcation control of a modified Pan-like chaotic system. Based on the Routh-Hurwtiz theory and high-dimensional Hopf bifurcation theory, the existence and stability of the Hopf bifurcation depending on selected values of the system parameters are studied. The region of the stability for the Hopf bifurcation is investigated.By the hybrid control method, a nonlinear controller is designed for changing the Hopf bifurcation point and expanding the range of the stability. Discussions show that with the change of parameters of the controller, the Hopf bifurcation emerges at an expected location with predicted properties and the range of the Hopf bifurcation stability is expanded. Finally,numerical simulation is provided to confirm the analytic results.     

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a newhyperchaotic circuit system. The stability of the hyperchaotic circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. Animportant feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.     

Convection in a weakly ionized plasma

The nonlinear dynamic stability of a weakly ionized plasma is investigated taking into account the spatial effects. Also obtained are the stability criteria.     

Conditional stability of diatomic molecule driven by a weak laser field

Using a direct perturbation method, we investigate the stability of a diatomic molecule modelled by a weakly laser-driven Morse oscillator. It is shown that stationary state solution of the system is stable in the sense of Lyapunov and the periodical one possesses conditional stability, namely its stability depends on the initial conditions and system parameters. The corresponding sufficient and necessary conditions are established that indicate the stable states associated with some discrete energies. The results reveal how a diatomic molecule can be stabilized or dissociated with a weak laser, and demonstrate that the mathematical conditional stability works in the considered physical system.     

Asymptotic analysis of a vibrating cantilever with a nonlinear boundary

Nonlinear vibration of a cantilever in a contact atomic force microscope is analyzed via an asymptotic approach. The asymptotic solution is sought for a beam equation with a nonlinear boundary condition. The steady-state responses are determined in primary resonance and subharmonic resonance. The relations between the response amplitudes and the excitation frequencies and amplitudes are derived from the solvability condition. Multivaluedness occurs in the relations as a consequence of the nonlinearity. The stability of steady-state responses is analyzed by use of the Lyapunov linearized stability theory. The stability analysis predicts the jumping phenomenon for certain parameters. The curves of the response amplitudes changing with the excitation frequencies are numerically compared with those obtained via the method of multiple scales. The calculation results demonstrate that the two methods predict the same varying tendencies while there are small quantitative differences. Supported by the National Outstanding Young Scientists Fund of China (Grant No. 10725209), the Shanghai Leading Academic Discipline Project (Grant No. S30106), and Shandong Jiaotong University Science Foundation (Grant No. Z200812) Recommended by LIAO ShiJun