Three-lobed shape bifurcation of rotating liquid drops

The evolution of axisymmetric equilibrium shapes of a rigidly rotating liquid drop can be extended beyond the 2-lobed shape bifurcation point if the rotating drop is driven in the n = 2 axisymmetric shape oscillation (perturbation), where n is the mode of oscillation. A reason for the extended stability of the perturbed rotating drop is that the inertia of the driven axisymmetric shape oscillation suppresses growth of a natural nonaxisymmetric shape fluctuation which leads to the 2-lobed shape bifurcation. The axisymmetric shape of the drop eventually bifurcates into either a 2- or a 3-lobed shape at a higher bifurcation point which is asserted to be the 3-lobed shape bifurcation point.     

The second Hopf bifurcation in lid-driven square cavity

To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of(11093.75, 11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs.     

Frequency locking by external force from a dynamical system with strange nonchaotic attractor

Usually, phase synchronization is studied in chaotic systems driven by either periodic force or chaotic force. In the present work, we consider frequency locking in chaotic Rössler oscillator by a special driving force from a dynamical system with a strange nonchaotic attractor. In this case, a transition from generalized marginal synchronization to frequency locking is observed. We investigate the bifurcation of the dynamical system and explain why generalized marginal synchronization can occur in this model.     

Delocalizing transition of Bose-Einstein condensates in optical lattices

A Bose-Einstein condensate trapped in a two-dimensional optical lattice exhibits an abrupt transition manifested by the macroscopic wave function changing character from spatially localized to extended. Resulting from a bifurcation, this irreversible transition takes place as the interwell potential barrier is adiabatically decreased below a critical value. This is in sharp contrast to the corresponding one-dimensional case where such a bifurcation is absent and the extent of a localized mode is continuously tunable. We demonstrate how these phenomena can be experimentally explored.     

负能模式在驱动漂移波非线性不稳定性中的作用(Ⅱ)——与正能模式交换,“回避交叉”和Hopf分岔

在正弦波驱动的非线性漂移波中,当驱动波频率和强度改变时,一个负能模式与另一个正能模式的复本征值可能出现交叉和“回避交叉”,在出现“回避交叉”的同时,这两个模式的地位发生了某种交换。正负能量两个模式之间的这种非线性共振在弱耗散下可能引起Hopf分岔。     

Numerical analysis of flame shape bifurcation in a two-stage swirled liquid burner using Large Eddy Simulation

A flame shape bifurcation in the liquid-fueled two-stage swirled BIMER combustor is studied using Large Eddy Simulations. This combustor, developed at the EM2C Laboratory to study Lean Premixed Prevaporized (LPP) burners, is composed of a two-stage injection system: a central swirled pilot stage fueled with a pressure-swirl atomizer, to sustain a piloting flame, and an outer swirled stage fed with a multi-point injection, to generate the LPP regime. After ignition in the pilot-only operating condition, a V flame is stabilized near the Inner Shear Layer (ISL). When switching to multipoint-only injection, a flame shape transition is observed and the flame bifurcates into a M-shape. In this work, we identify the mechanisms that lead to this bifurcation, and we show that the transition is driven by a complex coupling between the flame, the chamber acoustics and the ISL vortices. By switching to a multipoint-only injection, the fuel is essentially given to the ISL flame, which is mainly premixed. Because of the increased heat release rate and thanks to positive Rayleigh criterion, the quarter wave mode of the chamber is promoted. The ISL vortices, locked to this mode, increase in size until they are large enough to merge the flame in the CRZ, the radial momentum budget forcing the flow topology to switch to a bubble-like structure. Therefore, these results show that it is the existence of two possible flow topologies that renders this flame shape transition possible, the instability being responsible for transferring sufficient energy to the flow to enable the transitioning and the flame then changing its shape simply to adapt to the new topology.     

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：     

The onset of convection in a finite container due to surface tension and buoyancy

A method for predicting instabilities which combines recent techniques from bifurcation theory with the finite-element method is described. It is applied to the prediction of the onset of convection driven by both surface tension and buoyancy in rectangular containers. For zero buoyancy, the critical values of the Marangoni number for the first two bifurcations from the trivial solution are found for a two-dimensional cavity of aspect ratio 2. The variation of these critical values with aspect ratio is obtained by continuation methods and this reveals an interlacing of modes as the container size increases. It is established that the bifurcation to an even number of cells is transcritical rather than pitchfork and the turning point on the subcritical branch is located as a function of aspect ratio. The hysteresis associated with the transcritical bifurcation is small. As the surface tension forces decrease to zero, so that the convection is driven by buoyancy alone, the amount of hysteresis associated with the transcritical bifurcation becomes vanishingly small. The reason for this is not fully understood.     

Modulated amplitude waves and the transition from phase to defect chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures evolve towards defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.     

Critical exponent crossovers in escape near a bifurcation point

In periodically driven systems, near a bifurcation (critical) point the period-averaged escape rate Wmacr; scales with the field amplitude A as |ln(Wmacr;| proportional, variant (A(c)-A)(xi), where A(c) is a critical amplitude. We find three scaling regions. With increasing field frequency or decreasing |A(c)-A|, the critical exponent xi changes from xi=3/2 for a stationary system to a dynamical value xi=2 and then again to xi=3/2. Monte Carlo simulations agree with the scaling theory.     

非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.     

Nonequilibrium dynamical phase transition of 3D kinetic Ising／Heisenberg spin system

We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. T_{TCP}=1.33×exp(-ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H_{0}, reduced temperature T/T_C and the frequency ω as τ=C·ω^α·H_0^{-β}·(T/T_C)^{-γ}.     

Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

We study the stability and bifurcation of steady states for a certain kind of damped driven nonlinear Schrödinger equation with cubic nonlinearity and a detuning term in one space dimension, mathematically in a rigorous sense. It is known by numerical simulations that the system shows lots of coexisting spatially localized structures as a result of subcritical bifurcation. Since the equation does not have a variational structure, unlike the conservative case, we cannot apply a variational method for capturing the ground state. Hence, we analyze the equation from a viewpoint of bifurcation theory. In the case of a finite interval, we prove the fold bifurcation of nontrivial stationary solutions around the codimension two bifurcation point of the trivial equilibrium by exact computation of a fifth-order expansion on a center manifold reduction. In addition, we analyze the steady-state mode interaction and prove the bifurcation of mixed-mode solutions, which will be a germ of localized structures on a finite interval. Finally, we study the corresponding problem on the entire real line by use of spatial dynamics. We obtain a small dissipative soliton bifurcated adequately from the trivial equilibrium.     

负能模式在驱动漂移波非线性不稳定性中的作用(Ⅰ)——向正能模式的转变和双稳态

在正弦波驱动的非线性漂移波中,每一个模式都有与驱动波共振的参数区,驱动波能量的馈入引起共振负能模式向正能模式转变,这个转变与相应参数区中系统发展双稳态有关。     

基于麦克斯韦电磁场理论的神经元动力学响应与隐藏放电控制

钙、钾、钠等离子在细胞内连续泵送和传输时产生的时变电场不仅会影响神经元的放电活动,而且会诱导时变磁场去进一步调节细胞内离子的传播.根据麦克斯韦电磁场理论,时变的电场和磁场在细胞内外的电生理环境中会相互激发而产生电磁场.为了探究电磁场影响下的神经元放电节律转迁,本文在三维Hindmarsh-Rose(HR)神经元模型的基础上,引入磁通变量和电场变量,建立了一个五维HR神经元模型(简称EMFN模型).首先,结合Matcont软件分析了EMFN模型的平衡点分布与全局分岔性质,发现并分析了该模型存在的亚临界Hopf分岔、隐藏放电及其周期放电与静息态共存等现象.其次,利用双参数及单参数分岔、ISI分岔和最大Lyapunov指数等工具进行数值仿真,详细分析了EMFN模型存在的伴有混沌及无混沌的加周期分岔结构、混合模式放电和共存模式放电等现象,同时揭示了电场和磁场强度影响其放电节律的转迁规律.最后,利用Washout控制器将EMFN模型的亚临界Hopf分岔转化为超临界Hopf分岔,使其在分岔点附近的拓扑结构发生改变,由此达到消除其隐藏放电的目的.本文的研究结果证实了新建神经元模型具有丰富的放电节律,将影响神经元的信息传递和编码,为完善神经元模型,揭示电磁场对生物神经系统的影响,以及探求一些神经性疾病的致病机理提供了思路.     

Flow field topology of transient mixing driven by buoyancy

Transient mixing driven by buoyancy occurs through the birth of a symmetric Rayleigh-Taylor morphology (RTM) structure for large length scales. Beyond its critical bifurcation the RTM structure exhibits self-similarity and occurs on smaller and smaller length scales. The dynamics of the RTM structure, its nonlinear growth and internal collision, show that its genesis occurs from an explosive bifurcation which leads to the overlap of resonance regions in phase space. This event shows the coexistence of regular and chaotic regions in phase space which is corroborated with the existence of horseshoe maps. A measure of local chaos given by the topological entropy indicates that as the system evolves there is growth of uncertainty. Breakdown of the dissipative RTM structure occurs during the transition from explosive to catastrophic bifurcation; this event gives rise to annihilation of the separatrices which drives overlap of resonance regions. The global bifurcation of explosive and catastrophic events in phase space for the large length scale of the RTM structure serves as a template for which mixing occurs on smaller and smaller length scales.     

电流反馈型Buck变换器二维分段光滑系统边界碰撞和分岔研究

最近几年,边界碰撞分岔已经引起了越来越多的关注.以不连续导通模式下的电流反馈型Buck变换器为例,推导出两个边界三段形式的分段光滑系统的离散映射模型.数值仿真得到以参考电流为分岔参数的分岔图,然后具体分析定点的稳定存在域、分叉图中各段的映射构成和边界碰撞点处工作模式的转换.最后软件仿真和实验验证了二维分段光滑系统边界碰撞和分岔行为的正确性.     

Dynamics of transients in yttrium-iron-garnet

Yttrium-iron-garnet (YIG) is an important technological material used in microwave devices. In this paper we use dual microwave (1-4 GHz) drives to study the dynamical bifurcation behavior of magnetostatic and spin-wave modes in YIG spheres and rectangular films. The samples are placed in a dc magnetic field and driven by cw and pulse-modulated microwave excitations at magnetostatic mode frequencies. A second microwave drive applied to the sample excites additional spin-wave modes that can interact with those arising from the original excitation and thereby affect the transmission characteristics at the primary frequency. We find a significant decrease in transmission of the primary when the secondary frequency is tuned to approximately half that of the primary drive. This decrease is observed both in the steady state behavior and in the initial overshoot transient associated with pulse modulation of the primary excitation. Results such as these are often treated by extending linear theory to include higher order interaction terms. Herein we present a simple dynamical model that reproduces results that qualitatively resemble the experimental data. (c) 1997 American Institute of Physics.