Formation of a spiraling line defect and its meandering transition in a period-2 medium

The instability of a period-1 spiral wave resulting in a period-2 spiral wave with a line defect is investigated for the first time in a laboratory system. At the very onset the transition proceeds by an emergence of a spiraling line defect, "breathing" intermittently while retaining its symmetry of a period-1 spiral wave. With a further change in a control parameter, the line defect undergoes a meandering transition producing a compound tip trajectory, following a dynamic shape transition. The observed transitions have a strong analogy to the phase synchronization transition of two coupled nonlinear oscillators and the meandering transition of a period-1 spiral wave.     

Nucleation, drift, and decay of phase bubbles in period-2 oscillatory wave trains in a reaction-diffusion system

This is a report on experimental observations of phase bubbles, simply closed boundaries between domains oscillating 2pi out of phase, associated with period-2 oscillatory traveling waves in a Belousov-Zhabotinsky reaction-diffusion system. The bubbles nucleate spontaneously through a fast localized phase slip, drift radially away from a neighboring spiral wave core in an oscillatory fashion, and gradually shrink to disappear. Their oscillatory drift along the radial direction is a consequence of "period adaptation," while their lateral shrinkage is an attribute of local curvature. Similar dynamic structures can be reproduced in a simple, three-species reaction-diffusion model that supports period-2 oscillatory wave trains.     

Oscillations and chaos in CO+O(2) combustion

The gas-phase reaction between carbon monoxide and oxygen (in the presence of small amounts of hydrogen) shows bistability and oscillatory behavior. Typically, the oscillatory ignition has a period-1 relaxation waveform. The limit cycle is born at a saddle-node loop and terminates via a supercritical Hopf bifurcation. For a mean residence time of 8 s there is a period-doubling to a period-2 solution followed by period-halving to quasisinusoidal period-1 oscillations. At longer residence times, more period-doublings forming a full cascade to chaos with subsequent periodic windows are observed. The chaotic attractor has an underlying single-humped next maximum map.     

On the response of an oscillatory medium to defect generation

We investigate the response of a system far from equilibrium close to an oscillatory instability to the induction of phase singularities. We base our investigation on a numerical treatment of the complex Ginzburg–Landau equation (CGLE) in two spatial dimensions, which is considered as an order-parameter equation for lasers and other nonlinear optical systems. Defects are randomly generated by a spatially modulated linear growth rate. In the amplitude-turbulent regime, no qualitative change of behaviour can be detected. Phase-turbulent patterns emerging due to the Benjamin–Feir instability are destroyed by the externally injected defects. One observes either states consisting of spiral structures of various sizes which resemble the vortex glass states of the unperturbed system or a travelling wave pattern containing moving topological defects. In parameter space, both states are separated by a well-defined phase boundary which is close to the line separating convectively from absolutely stable travelling waves. PACS 47.54.+r; 89.75.Kd; 42.65.Sf; 47.32.Cc; 47.27.Cn; 05.45.-a     

反应扩散系统中螺旋波的失稳

文章以反应扩散系统为例,介绍了在可激发系统与振荡系统中螺旋波产生、发展、演化的一些基本性质及规律,并讨论了作者近年来对螺旋波的各种失稳途径、时空混沌的产生机理及螺旋波控制方面所做的实验与理论工作,重点讨论了两类螺旋波失稳现象：爱克豪斯失稳与多普勒失稳,两类失稳都使系统从有规律的螺旋波态变为时空混沌（缺陷湍流）态。     

Nonlinear waves in nonequilibrium systems of the oscillatory type,part I

Nonlinear waves in mathematical models of nonequilibrium spatially uniform media with the oscillatory instability of the trivial state are considered. The models are based on the generalized Ginsburg-Landau equations. For the long-wave system, i.e. that described by two-component reaction-diffusion equations, we obtain the full stability conditions for monochromatic plane travelling waves. The basic part of the paper is devoted to the short-wave system which can be described by reaction-diffusion equations with not less than three components or by a two-component system with residual nonlocality. We construct the Ginsburg-Landau equation for this system, and we find its general quasistationary one-dimensional solution which is a travelling wave modulated by a travelling envelope wave. The stability of this solution is investigated with the especial emphasis on different important particular cases. The obtained results are compared with experimental observations of different waves on fronts of detonation and non-gaseous combustion (which also are characterized by the oscillatory short-wave instability of the trivial state), and the qualitative agreement between theoretical and experimental results is demonstrated.     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

Structure and dynamics of the Frenkel-Kontorova dislocations in electroconvection in liquid crystals

The Frenkel-Kontorova instability is studied in a 1D lattice of domains formed during electroconvection in nematic liquid crystals twisted by π/2. It is found that generation of defects by such instability can be observed in this model medium. Among other things, it is shown that several types of defects with singular and nonsingular cores, as well as with a extended core, are formed in the 1D domain structure above the electroconvective instability threshold. The extended cores of dislocations are dissociated into a line, and the entire structure is isomorphic to two partial dislocations spaced by a certain distance, which are not observed in free form. Defects with a nonsingular core (zero topological index) exist owing to spiral hydrodynamic flows in convective rolls and are not observed in layers with a homogeneous orientation of molecules. It is shown that the formation of both types of defects follows the scenario of decay of dislocations with extended cores via detachment of nonsingular defects (i.e., discretely); as a result, a dislocation with a singular core is left. “Breather” defects, which are the result of periodic creation and annihilation of dislocations with a topological index of ±1, are also observed. The effect of defects on the transition from the 1D to 2D structures is considered.     

Stability of oscillatory flows past compliant surfaces

The stability of oscillatory flows over compliant surfaces is studied analytically and numerically. The types of compliant surfaces studied are the spring backed wall model, which permits tangential motion of the surface, and the incompressible viscoelastic gel model. The stability is determined using the Floquet analysis, where amplitude of perturbations at time intervals separated by one time period is examined to determine whether perturbations grow or decay. The oscillatory flows past both the spring backed wall model and the viscoelastic gel model exhibit an instability in the limit of zero Reynolds number, and the transition amplitude of the oscillatory velocity increases with the frequency of oscillations. The transition amplitude has a minimum at zero wave number for the spring backed plate model, whereas the minimum occurs at finite wavenumber for the viscoelastic gel model. For the spring backed plate model, it is shown that the instability due to steady mean flow and the purely oscillatory instability reinforce each other, and the regions of instability are mapped in the ( ) plane, where is the steady strain rate and A is the oscillatory strain rate. For the viscoelastic gel model, the instability is found to depend strongly on the gel viscosity , and the effect of oscillations on the continuation of viscous modes at intermediate Reynolds number shows a complicated dependence on the oscillation frequency.Received: 17 March 2004, Published online: 30 September 2004PACS: 47.20.Ft Instability of shear flows - 83.50.-v Deformation and flow - 87.19.Tt Rheology of body fluids     

Complex transitions between spike,burst or chaos synchronization states in coupled neurons with coexisting bursting patterns

We investigated the synchronization dynamics of a coupled neuronal system composed of two identical Chay model neurons. The Chay model showed coexisting period-1 and period-2 bursting patterns as a parameter and initial values are varied. We simulated multiple periodic and chaotic bursting patterns with non-(NS), burst phase(BS), spike phase(SS),complete(CS), and lag synchronization states. When the coexisting behavior is near period-2 bursting, the transitions of synchronization states of the coupled system follows very complex transitions that begins with transitions between BS and SS, moves to transitions between CS and SS, and to CS. Most initial values lead to the CS state of period-2 bursting while only a few lead to the CS state of period-1 bursting. When the coexisting behavior is near period-1 bursting, the transitions begin with NS, move to transitions between SS and BS, to transitions between SS and CS, and then to CS. Most initial values lead to the CS state of period-1 bursting but a few lead to the CS state of period-2 bursting. The BS was identified as chaos synchronization. The patterns for NS and transitions between BS and SS are insensitive to initial values. The patterns for transitions between CS and SS and the CS state are sensitive to them. The number of spikes per burst of non-CS bursting increases with increasing coupling strength. These results not only reveal the initial value- and parameterdependent synchronization transitions of coupled systems with coexisting behaviors, but also facilitate interpretation of various bursting patterns and synchronization transitions generated in the nervous system with weak coupling strength.     

各向异性表面张力对深胞晶界面形态稳定性的影响

应用匹配渐近展开法和多变量展开法研究各向异性表面张力对定向凝固中深胞晶界面形态稳定性的影响, 通过寻找定向凝固系统的模式解获得了深胞晶界面形态满足的量子化条件. 结果表明, 与各向同性的定向凝固系统中深胞晶界面形态稳定性比较, 考虑各向异性表面张力的定向凝固中深胞晶生长界面形态也有两种整体不稳定性机制: 整体波动不稳定性和低频不稳定性. 随着各向异性表面张力的增加, 中性模式产生强振荡的枝晶结构的整体波动不稳定性的不稳定区域减小, 中性模式产生弱振荡的胞晶结构的低频不稳定性的不稳定区域增加.     

Co-existence of multiple attractors in the PWM controlled DC drives

DC shunt and series drives are extensively used in the industry. The occurrence of bifurcation and chaos in dc shunt and permanent magnet drives are well known. It is observed that the behavior of the drives not only depends on the value of system parameters but also on the value of initial conditions. Multiple attractors can exist for same parameter value. Different choice of initial conditions gives different periodic behavior of the system. The drive is intended to operate in a parameter range to give period-1 behavior. We report the existence of sub- harmonic oscillations in the period-1 region of the bifurcation diagram along with co-existing attractor with fractal basin boundaries in PWM controlled dc series drives. The series drive is extensively used in electric traction and other applications. The dc drives are run with dc input voltage. This dc voltage may be derived from a dc source or an ac source with a rectifier. The dc series drive shows different bifurcation behavior when different types of input voltage and switching elements are used. The existence of period-1, period-2 and period-4 orbits are observed with different initial conditions in the desired period-1 region of the bifurcation diagram. The dependence of system’s behavior on initial condition may render the system’s behavior unpredictable. These phenomena may have serious implication in performance.     

Bursting and spiking due to additional direct and stochastic currents in neuron models

Neurons at rest can exhibit diverse firing activities patterns in response to various external deterministic and random stimuli, especially additional currents. In this paper, neuronal firing patterns from bursting to spiking, induced by additional direct and stochastic currents, are explored in rest states corresponding to two values of the parameter $V_{\rm K}$ in the Chay neuron system. Three cases are considered by numerical simulation and fast/slow dynamic analysis, in which only the direct current or the stochastic current exists, or the direct and stochastic currents coexist. Meanwhile, several important bursting patterns in neuronal experiments, such as the period-1 ``circle/homoclinic" bursting and the integer multiple ``fold/homoclinic" bursting with one spike per burst, as well as the transition from integer multiple bursting to period-1 ``circle/homoclinic" bursting and that from stochastic ``Hopf/homoclinic" bursting to ``Hopf/homoclinic" bursting, are investigated in detail.     

State-to-State Transitions in a Hindmarsh-Rose Neuron System

We investigate the dynamical response of the neuron system to a feeble external signal by using the Hindmarsh-Rose model, when the system is tuned below the first bifurcation point, which corresponds to the period-1 bursting state, and an external signal with a fixed period of about 170s is introduced to the system. It is found that to respond to the outside signal, the system changes from the period-1 state to a period-2 one with variation of the signal amplitude, indicating the occurrence of state-to-state transition （SST）. Moreover, when a signal with different fixed periods is introduced, we can also find a similar transition between other states. Furthermore, the effect of the frequency of the signal on the transition is also discussed. These results may imply that SST plays a constructive role in information processing in neuron systems.     

Formation of defect density superlattices in binary compounds under nuclear irradiation

Manifestation of antisite defects in crystals under irradiation is studied. Calculations show that the concentration of such defects can reach high levels under typical irradiation intensities and temperatures. It is shown that buildup of antisite defects and interaction between them can result in instability of the system during irradiation with respect to spatially nonuniform perturbations. The instability should give rise to a periodic modulation of the density of antisite defects. The region of instability as a function of crystal parameters and irradiation has been obtained.     

谷值V~2控制Boost变换器的精确建模与动力学分析

建立了谷值V2控制Boost变换器的离散迭代映射模型,在此基础上得到了输入电压、输出电容及其等效串联电阻(equivalent series resistance,ESR)变化时的分岔图,推导了不动点处的雅可比矩阵,利用特征值和最大Lyapunov指数对系统进行了稳定性分析,并验证了分岔图的正确性.重点研究了输入电压和输出电容及其ESR对谷值V2控制Boost变换器的动力学特性的影响.研究结果表明,输入电压增大时,变换器从周期1态经历1次倍周期分岔和边界碰撞分岔进入混沌状态;输出电容及其ESR具有相同的分岔路由,随着输出电容及其ESR的逐渐减小,变换器具有从周期1态经历周期2态、周期4态、周期8态、逐渐演变到混沌态的动力学行为.最后,用仿真和实验结果验证了本文理论分析的正确性.     

Experimental study of small signal amplification in Bragg acoustooptic bistable system

The external small signal amplification ability of the Bragg acoustooptic system has been studied in this paper. It has been proven by experiment that there are saturation and resonance phenomena in the amplification. Bifurcation parameters at the bifurcation points are decreased by external simple harmonic signals. At the period-2 bifurcation point, small signal amplification energy is mainly from the period-2 component. External signals have the ability of frequency pulling and synchronising period-2 frequency. These phenomena have great significance concerning bifurcation and chaos.     

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.     

Pre-B?tzinger复合体的从簇到峰放电的同步转迁及分岔机制

Pre-Bötzinger复合体是兴奋性耦合的神经元网络,通过产生复杂的放电节律和节律模式的同步转迁参与调控呼吸节律.本文选用复杂簇和峰放电节律的单神经元数学模型构建复合体模型,仿真了与生物学实验相关的多类同步节律模式及其复杂转迁历程,并利用快慢变量分离揭示了相应的分岔机制.当初值相同时,随着兴奋性耦合强度的增加,复合体模型依次表现出完全同步的“fold/homoclinic”,“subHopf/subHopf”簇放电和周期1峰放电.当初值不同时,随耦合强度增加,表现为由“fold/homoclinic”,到“fold/fold limit cycle”、到“subHopf/subHopf”与“fold/fold limit cycle”的混合簇放电、再到“subHopf/subHopf”簇放电的相位同步转迁,最后到反相同步周期1峰放电.完全(同相)同步和反相同步的周期1节律表现出了不同分岔机制.反相峰同步行为给出了与强兴奋性耦合容易诱发同相同步这一传统观念不同的新示例.研究结果给出了preBötzinger复合体的从簇到峰放电节律的同步转迁规律及复杂分岔机制,反常同步行为丰富了非线性动力学的内涵.     

Spatiotemporal oscillations and rheochaos in a simple model of shear banding

We study a simple model of shear banding in which the flow-induced phase is destabilized by coupling between flow and microstructure (wormlike micellar length). By varying the strength of instability and the applied shear rate, we find a rich variety of oscillatory and chaotic shear banded flows. At low shear and weak instability, the induced phase pulsates next to one wall of the flow cell. For stronger instability, high shear pulses ricochet across the cell. At high shear we see oscillating bands on either side of central defects. We discuss our results in the context of recent experiments.