Transverse instability of line defects of period-2 spiral waves

Spiral waves that arise in period-2 oscillatory media extended in space generically bear "line defects" along which the local kinetics exhibits a period-1 oscillation. Locally, these defect structures can be viewed as a front separating two period-2 oscillatory domains oscillating 2pi out of phase. Here we show that their shape can become sinusoidal with a transverse instability as in bistable fronts. This instability eventually leads to a line-defect filled spatiotemporal chaotic state having erratic proliferations, annihilations, and regenerations of line defects. The same sequence of phenomena is observed in a model reaction-diffusion system as well as in an experimental system.     

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.     

一类新的混沌神经放电的动力学特征的实验和数学模型研究

神经元电活动理论模型Hindmarsh-Rose(HR)模型提示有位于周期1和周期2放电模式之间的一类特殊的混沌放电,但长期以来对其没有获得足够认识.依据回归映射的确定性结构和非线性预报的短期可预报性,确认了在大鼠的实验性神经起步点的实验中发现的位于周期1和周期2放电模式之间的非周期放电是混沌放电模式,还将该混沌放电模式区分为3个不同表观样式.其中1个表观形式与HR模型的仿真结果相类似,验证了HR模型的理论预期;其余2个样式与仿真结果并不相似.进一步揭示了3个表观样式的动力学特征以及相互之间的区别与联系,并与位于周期2和周期3节律之间、周期3和周期4节律之间的混沌比较了异同,也区别了从周期1到混沌再到周期2放电模式的节律转迁历程与其他的从周期1到周期2节律的分岔过程的不同.研究结果确认了该类特殊混沌节律和相应分岔过程的新特征,丰富了混沌放电节律和节律分岔序列的种类.还对仿真该混沌的多样性和非光滑特性,以及揭示该类混沌的产生途径等进行了讨论. 关键词： 混沌 神经放电模式 分岔 节律     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Sampled-data modeling and dynamical effect of output-capacitor time-constant for valley voltage-mode controlled ewline buck-boost converter

By analyzing the output voltage ripple of a buck-boost converter with large equivalent series resistance (ESR) of output capacitor, one valley voltage-mode controller for buck-boost converter is proposed. Considering the fact that the increasing and decreasing slopes of the inductor current are assumed to be constant during each switching cycle, an especial sampled-data model of valley voltage-mode controlled buck-boost converter is established. Based on this model, the dynamical effect of an output-capacitor time-constant on the valley voltage-mode controlled buck-boost converter is revealed and analyzed via the bifurcation diagrams, the movements of eigenvalues, the Lyapunov exponent spectra, the boundary equations, and the operating-state regions. It is found that with gradual reduction of output-capacitor time-constant, the buck-boost converter in continuous conduction mode (CCM) shows the evolutive dynamic behavior from period-1 to period-2, period-4, period-8, chaos, and invalid state. The stability boundary and the invalidated boundary are derived theoretically by stability analysis, where the stable state of valley voltage-mode controlled buck-boost converter can enter into an unstable state, and the converter can shift from the operation region to a forbidden region. These results verified by time-domain waveforms and phase portraits of both simulation and experiment indicate that the sampled-data model is correct and the time constant of the output capacitor is a critical factor for valley voltage-mode controlled buck-boost converter, which has a significant effect on the dynamics as well as control stability.     

Nonlinear resonance phenomena of a doped fibre laser under cavity-loss modulation: Experimental demonstrations

Our experiments with an erbium-doped fibre ring laser (CW, single transverse mode and multiaxial mode) with an intracavity LiNbO₃ electro-optic modulator (EOM ) display the characteristic features of a nonlinear oscillator (e.g., harmonic and period-2 sub-harmonic resonances ) when the EOM driver voltage is modulated periodically. Harmonic resonance leads to period-1 bistability and hysteresis. Inside the period-2 sub-harmonic resonance region, the laser exhibits Feigenbaum sequence and generalized bistability.     

Alternans and higher-order rhythms in an ionic model of a sheet of ischemic ventricular muscle

Life-threatening arrhythmias such as ventricular tachycardia and fibrillation often occur during acute myocardial ischemia. During the first few minutes following coronary occlusion, there is a gradual rise in the extracellular concentration of potassium ions ([K(+)](0)) within ischemic tissue. This elevation of [K(+)](0) is one of the main causes of the electrophysiological changes produced by ischemia, and has been implicated in inducing arrhythmias. We investigate an ionic model of a 3 cmx3 cm sheet of normal ventricular myocardium containing an ischemic zone, simulated by elevating [K(+)](0) within a centrally-placed 1 cmx1 cm area of the sheet. As [K(+)](0) is gradually raised within the ischemic zone from the normal value of 5.4 mM, conduction first slows within the ischemic zone and then, at higher [K(+)](0), an arc of block develops within that area. The area distal to the arc of block is activated in a delayed fashion by a retrogradely moving wavefront originating from the distal edge of the ischemic zone. With a further increase in [K(+)](0), the point eventually comes where a very small increase in [K(+)](0) (0.01 mM) results in the abrupt transition from a global period-1 rhythm to a global period-2 rhythm in the sheet. In the peripheral part of the ischemic zone and in the normal area surrounding it, there is an alternation of action potential duration, producing a 2:2 response. Within the core of the ischemic zone, there is an alternation between an action potential and a maintained small-amplitude response ( approximately 30 mV in height). With a further increase of [K(+)](0), the maintained small-amplitude response turns into a decrementing subthreshold response, so that there is 2:1 block in the central part of the ischemic zone. A still further increase of [K(+)](0) leads to a transition in the sheet from a global period-2 to a period-4 rhythm, and then to period-6 and period-8 rhythms, and finally to a complete block of propagation within the ischemic core. When the size of the sheet is increased to 4 cmx4 cm (with a 2 cmx2 cm ischemic area), one observes essentially the same sequence of rhythms, except that the period-6 rhythm is not seen. Very similar sequences of rhythms are seen as [K(+)](0) is increased in the central region (1 or 2 cm long) of a thin strand of tissue (3 or 4 cm long) in which propagation is essentially one-dimensional and in which retrograde propagation does not occur. While reentrant rhythms resembling tachycardia and fibrillation were not encountered in the above simulations, well-known precursors to such rhythms (e.g., delayed activation, arcs of block, two-component upstrokes, retrograde activation, nascent spiral tips, alternans) were seen. We outline how additional modifications to the ischemic model might result in the emergence of reentrant rhythms following alternans. (c) 2000 American Institute of Physics.     

Sampled-data modeling and dynamical effect of output-capacitor time-constant for valley voltage-mode controlled buck-boost converter

By analyzing the output voltage ripple of a buck-boost converter with large equivalent series resistance(ESR) of output capacitor, one valley voltage-mode controller for buck-boost converter is proposed. Considering the fact that the increasing and decreasing slopes of the inductor current are assumed to be constant during each switching cycle, an especial sampleddata model of valley voltage-mode controlled buck-boost converter is established. Based on this model, the dynamical effect of an output-capacitor time-constant on the valley voltage-mode controlled buck-boost converter is revealed and analyzed via the bifurcation diagrams, the movements of eigenvalues, the Lyapunov exponent spectra, the boundary equations,and the operating-state regions. It is found that with gradual reduction of output-capacitor time-constant, the buck-boost converter in continuous conduction mode(CCM) shows the evolutive dynamic behavior from period-1 to period-2, period-4, period-8, chaos, and invalid state. The stability boundary and the invalidated boundary are derived theoretically by stability analysis, where the stable state of valley voltage-mode controlled buck-boost converter can enter into an unstable state, and the converter can shift from the operation region to a forbidden region. These results verified by time-domain waveforms and phase portraits of both simulation and experiment indicate that the sampled-data model is correct and the time constant of the output capacitor is a critical factor for valley voltage-mode controlled buck-boost converter, which has a significant effect on the dynamics as well as control stability.     

电压型双频率控制开关变换器的动力学建模与多周期行为分析

在断续导电模式下, 建立了电压型双频率控制开关变换器的动力学模型, 并推导了相应的特征值方程. 根据动力学模型, 采用分岔图研究了电路参数变化时变换器存在的边界碰撞分岔行为和周期2, 周期3,周期4等多周期行为, 结果表明: 变换器经历了周期1态、多周期态、周期1态的分岔路由; 周期态的转变均是由边界碰撞分岔引起的. 根据特征值方程, 采用Lyapunov指数研究了变换器的稳定性, 结果表明: 随着电路参数的变化, Lyapunov指数始终小于零, 变换器一直工作于稳定的周期态, 验证了电压型双频率控制开关变换器的周期3行为并不意味着变换器会必然发生混沌. 通过电路仿真, 分析了负载变化时变换器的时域波形、相轨图和频谱图, 验证了动力学模型的可行性和理论分析的正确性. 实验结果验证了文中的仿真结果. 关键词： 开关变换器 双频率控制 边界碰撞分岔 多周期行为     

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.     

Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.     

The use of nonlinear dynamics of erbium-doped fiber laser at pump modulation for intra-cavity sensing

We report the use of nonlinear dynamics of an erbium-doped fiber laser (EDFL) at pump modulation, namely the period-1 pulsed regime, for intra-cavity loss sensing and demonstrate noticeable sensitivity enhancement of an EDFL-based sensor, provided by this arrangement. Experimentally, we obtain a ten-fold increase of the sensor response, i.e. peak-to-peak pulse amplitude against loss variation, as compared with the standard sensing schemes using a low-power wide-spectrum light emitting diode (LED) or fiber laser without external modulation. This advantage originates from a strong dependence of pulse amplitude in the period-1 regime on intra-cavity loss variable that is, in turn, determined by the interrelation of relaxation frequency (an internal EDFL parameter) and frequency and depth of external (pump) modulation. A modeling of the EDFL-based sensor, presented for the case when the laser operates in the period-1 regime, allows an insight into the sensor operation details and opens the gate to its further optimization. The proposed sensing method seems to be a proper choice for the applications where an intra-cavity sensor’s head has high internal loss (≥10 dB) while sensed loss is varied within a quite narrow range (a few dB).     

The dynamics of a symmetric coupling of three modified quadratic maps

We investigate the dynamical behavior of a symmetric linear coupling of three quadratic maps with exponential terms, and identify various interesting features as a function of two control parameters. In particular, we investigate the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable period-l, period-2, and period-3 orbits. We also investigate the multistability in the same coupling. Lyapunov exponents, parameter planes, phase space portraits, and bifurcation diagrams are used to investigate transitions from periodic to quasiperiodic states, from quasiperiodic to mode-locked states and to chaotic states, and from chaotic to hyperchaotic states.     

蔡氏电路的研究与制作

在非线性电路中蔡氏电路是迄今为止产生复杂动力学行为的最为有效和较为简单的电路之一。从软件仿真和电路制作两个角度分别对蔡氏电路的特性进行了研究,讨论蔡氏电路的工作原理,分别用PSPICE和MATLAB软件仿真出蔡氏电路的相关特性,并制作出硬件电路,给出观察蔡氏电路周期1、周期2极限环及单涡卷和双涡卷混沌吸引子的实验结果,得出了与理论分析一致的结果。对进一步研究和设计专用混沌集成电路及混沌在保密通信中的应用具有积极的意义。