Brown方程的分歧解与初始成畴问题

初始成畴问题是一个典型的分歧问题。通过标度变换使铁磁体的尺寸参数显现在自由能表达式和Brown方程中。推广应用分歧理论的微扰法,求出了Brown方程的最小正分歧点和从此点发出的分歧解。分析了单畴态和分歧解的稳定性,据此对初始成畴临界尺寸和初始成畴行为进行了研究。结论指出,临界尺寸是尺寸参数的最小正分歧点,其近旁的初始成畴过程或者沿着分歧解连续地进行,或者是一个不连续的跃变。给出了相应的判据。此外,还得到了严格单畴临界尺寸上下限的一个估计,以及在特殊情况下这种临界尺寸的精确结果。 关键词：     

Stability of a detuned single mode homogeneously broadened ring laser

We study analytically the influence of detuning on the properties of a homogeneously broadened single mode ring laser. In the good cavity domain the stationary output is stable. In the bad cavity domain the stationary output becomes unstable via a Hopf bifurcation. Near line center this Hopf bifurcation is subcritical, leading to unstable small amplitude oscillations. Far from line center this Hopf bifurcation is supercritical and leads to stable small amplitude output.     

Bifurcation phenomena in a laser with saturable absorber I

We investigate the bifurcation diagram of a laser with saturable absorber in the low and medium intensity regimes. The linear stability of the stationary solutions corresponding to these regimes is studied. In the low intensity domain, a Hopf bifurcation point is determined from which a time-periodic solution emerges. This solution is contructed and its stability is analyzed in the vicinity of the bifurcation point. It is shown that this time-periodic solution is stable in a finite domain of the parameter space.     

A new type of global bifurcation in Hénon map

A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n≥1) solution, and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram, and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the Hénon map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m≥3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged, and there is a crisis step near the landing area.     

Nondegenerate electron solids

The transition from a nondegenerate electron gas to an “electron solid” is described as a bifurcation. A trial function calculation is performed to determine the bifurcation domain in the plane of mean interelectron spacing and temperature.     

时滞惯性神经网络的稳定性和分岔控制

针对一类二阶时滞惯性神经网络模型,提出一种基于时滞反馈的分岔控制方法.利用时滞微分方程动力学理论,给出反馈控制系统的稳定性以及发生Hopf分岔的判别条件.数值仿真显示所设计的控制器不仅能有效延迟网络分岔的发生,还能扩大稳定域并改善网络的收敛速度.     

一类非线性相对转动系统的组合谐波分岔行为研究

针对一类具有非线性刚度、非线性阻尼的非线性相对转动系统, 应用耗散系统的拉格朗日原理建立在组合谐波激励作用下非线性相对转动系统的动力学方程. 构造李雅普诺夫函数, 分析相对转动系统的稳定性, 研究自治系统的分岔特性. 应用多尺度法求解相对转动系统的非自治系统在组合激励作用下的分岔响应方程. 最后采用数值仿真方法, 通过分岔图、时域波形、相平面图、Poincaré截面图等研究外扰激励、系统阻尼、 非线性刚度对相对转动系统经历倍周期分岔进入混沌运动的影响. 关键词： 相对转动 组合激励 分岔 混沌     

Global control of spiral wave dynamics in an excitable domain of circular and elliptical shape

Experiments performed in a thin layer of the Belousov-Zhabotinsky solution subjected to a global feedback demonstrate the existence of the resonance attractor for meandering spiral waves within a domain of circular shape. In an elliptical domain, the resonance attractor can be destroyed due to a saddle-node bifurcation induced by a variation of the domain eccentricity. This conclusion explains the experimentally observed anchoring of spiral waves at certain points of an elliptical domain and is in good quantitative agreement with numerical data obtained for the Oregonator model.     

Response of nonlinear random business cycle model with time delay state feedback

The steady state response and bifurcation of nonlinear random business cycle model to random narrow-band excitation with time delay state feedback are studied in this paper. The method of multiple scales is used to determine the business cycle model of modulation of amplitude and phase. The effects of delay, detuning, bandwidth and magnitude of random excitation on dynamics of the business cycle system are investigated. The results show that the complex dynamics such as bifurcation, jump domain and so on are induced by time delay and the phenomena that multiple solution or bifurcation is induced by noise.     

On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.     

The magnetic domain structure of partially crystallized amorphous Fe—B alloys by electron microscopy

A detailed electron microscopy study has been made of the crystallization behaviour up to 500°C of some amorphous FeB alloys together with accompanying changes in domain structure. Across the whole composition range investigated, crystallization involves the creation of small grains of bcc α-iron as well as larger spherulites comprised mostly of Fe₃B. Both stages of the crystallization process produce remarkable changes in the domain structure, e.g. the formation of magnetization ripple, which to some extent are composition-dependent. These changes are discussed in some detail and models proposed to explain them.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Mode selection in nearly-self-sustained explosive crystallization

We study a model of a driven thermally activated, exothermic process, in the context of explosive crystallization of a thin amorphous film. In the limit in which self-sustained crystallization at a speed close to the laser scan speed is almost possible, the model reduces to a single autonomous second-order differential equation for the amorphous-crystalline interface position. The steady state of this equation represents a straight-line interface advancing into the amorphous region at a speed equal to the laser scan speed. As a control parameter is increased, this steady state of the model becomes linearly unstable, undergoing a Hopf bifurcation which gives rise to oscillations of the interface velocity. For parameter ranges in which the interface remains a straight line, the amplitude, period, and form of these oscillations is calculated as a function of the control parameter. A similar equation, valid when the interface does not necessarily remain straight but may take on a long-wavelength shape, is also derived; it is harder to analyze.     

Consequences of using different pair-correlation functions on the stability properties of the Homogeneous Cooling State for a monodisperse system of near-elastic disks

We show the differences in the stability properties of the Homogeneous Cooling State (HCS) of a two-dimensional monodisperse collection of rigid and near-elastic disks, obtained by using different formulae for the pair-correlation function.For an equation of state that takes into account the crystallization and ordering of the particles (and the respective pressure drop), the critical wavelength of the heat conduction mode is considerably modified in the transition zone, involving a bifurcation and an additional mode of instability. The theoretical predictions, using the improved equation of state are confirmed by numerical simulations. Nevertheless, some open questions remain.     

Coexistence of single-mode and multi-longitudinal mode emission in the ring laser model

A homogeneously broadened unidirectional ring laser can emit in several longitudinal modes for large enough pump and cavity length because of Rabi splitting induced gain. This is the so called Risken-Nummedal-Graham-Haken (RNGH) instability. We investigate numerically the properties of the multi-mode solution. We show that this solution can coexist with the single-mode one, and its stability domain can extend to pump values smaller than the critical pump of the RNGH instability. Moreover, we show that the multi-mode solution for large pump values is affected by two different instabilities: a pitchfork bifurcation, which preserves phase-locking, and a Hopf bifurcation, which destroys it.     

Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation

We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states(QSs) and the repetitive spiking states(SPs) may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.     

Influence of Polycarbonate on the Crystallization Behavior of Poly(Trimethylene Terephthalate)/Polycarbonate Blends

To determine the factors influencing the retardation of the crystallization of poly(trimethylene terephthalate) (PTT) when PTT is blended with polycarbonate (PC), different PTT/PC blends were prepared via the melt mixing method. The relationships between the crystallization behavior and blend composition, as well as the phase morphology, were investigated. The results showed that the predominant reason for the retardation in crystallization is due to the PC content and phase morphology. The PC influences the crystallization of PTT via two methods. First, it retards PTT crystallization. Secondly, the PC exhibits a nucleation effect on the PTT crystallization which is, however, much weaker compared to the negative effect PC exerts with regards to PTT crystallization. When the processing temperature and shear rate remains unchanged, the two effects of PC determine the crystallization behavior of the blend. The phase morphology, which is strongly dependent on the mixing temperature and the shear rate, and which is also related to mixing time, had an appreciable impact on PTT crystallization. In the case of similar adhesion with the interface, a finer PC phase domain would show a slightly stronger nucleation effect on PTT crystallization.     

Power spectra of active galactic nuclei

The power spectral densities (PSDs) for a sample of active galactic nuclei (AGNs) are analyzed in both the frequency domain and the time domain. We find for each object that for broadband noise a character timescale-bifurcation timescale of Fourier and time-domain PSD exists in the 10³–10⁶ s range, below which the time-domain power spectrum is systematically higher than the corresponding Fourier spectrum. The relationship between bifurcation timescale, AGN mass and luminosity is studied. Compared with the fact that similar phenomena have been found for Galactic black hole candidates (GBHs) with bifurcation timescale ∼0.1 s but not for accreting neutron stars, our finding indicates that AGNs and GBHs have common intrinsic nature in rapid X-ray variability with a character time parameter scaled with their masses Supported by the Special Funds for Major State Basic Research Projects (Grant No. 2009CB824800) and the National Natural Science Foundation of China (Grant No. 10533020)     

Asymptotics of large bound states of localized structures

We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods. We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.     

Effect of nonlinear translations on the pulsation of cavitation bubbles

The pulsations and translations of cavitation bubbles obey combined ordinary differential equations, and their nonlinearities are studied by the bifurcation diagram and the phase diagram in a strong ultrasonic field. Bubble pulsation can change regularly or irregularly with changing driving pressure in the time domain. The bifurcation diagrams of the pulsation versus driving pressure show that the pulsations and translations of bubbles have nonlinear characteristics, and the nonlinear translations of bubbles can disorder the pulsations for certain parameters. Disorder of the pulsation can also be caused by nonlinear pulsation itself. In addition, the phase diagrams also show that the nonlinear translations make a large contribution to the pulsations. The same result can also be obtained when the ambient radii of two bubbles are different.