一种“类耗散系统”中的“类Ⅴ型阵发”

一类不连续不可逆保面积映象可以展示类似耗散的行为,因此可称其为“类耗散系统”.在一种类耗散系统中观察到了椭圆周期轨道及其周围的椭圆岛与映象不连续边界碰撞而消失的现象.周期轨道消失后,经过一系列过渡椭圆周期轨道之后,系统的行为由一个混沌类吸引子主导.在混沌类吸引子刚刚出现时,混沌时间序列呈现层流相与湍流相的无规交替.这一切都与不连续耗散系统中发生的Ⅴ型阵发的相应性质十分相似,因此可称为“类Ⅴ型阵发”.然而,当混沌类吸引子刚刚出现时,仅可以找到最后一个过渡椭圆岛的“遗迹”,并不存在它的“鬼魂”,因此类Ⅴ型阵发不遵从Ⅴ型阵发的特征标度规律.反之,混沌类吸引子的鬼魂却存在于最后一个过渡椭圆周期轨道的类瞬态过程中,因此在类Ⅴ型阵发导致混沌运动的临界点之前,由此“类瞬态混沌奇异集”中逃逸的规律就成为标志这一种临界现象的标度律.这与Ⅴ型阵发又根本不同. 关键词： 类耗散性 类混沌吸引子 类Ⅴ型阵发     

若干二维映象中V型阵发层流相区的特征

通过几个实例的分析,说明二维映象中V型阵发的层流相所占据的相空间区域常常具有一维特征.这些一维相空间区域对应于已经由边界碰撞分岔消失的周期点的“鬼魂”留下的一段稳定流形.它们构成类似于一维映象中阵发发生后层流相迭代通过的“隧道”.隧道的开口有明确的位置.V型阵发的湍流相所对应相空间区域具有二维特征.解析讨论了这些特征的形成机制. 关键词： V型阵发 层流相 流形     

截断误差诱导阵发混沌与拓展维

进一步研究过焦系统，找到了截断误差诱导阵发混沌的直接实验证据.实验显示，过焦系统的数字解呈现开关阵发混沌的特征，分析解仅是一个简单极限序列；影子效应支配系统演化；截断误差的累计误差是演化轨道的影子距离，影子距离随系统逼近不动点增加，随系统远离不动点减少，增加量与减少量统计上相等；在非双曲不动点的邻域内，存在一个阈值，当累计误差超过该阈值时，影子效应失灵，系统从吸引相跳到排斥相.一种合理的猜测是在非双曲不动点局域截断误差可以诱导出新维度——拓展维. 关键词： 开关阵发混沌 截断误差 过焦系统 拓展维     

一个光学系统展示的V型阵发现象

本文报道基于一个光学系统的延时微分方程和方程的Poincaré映象计算得到的V型阵发及其主要标度规律,为我国科学工作者提出的阵发新型提供一个新的证据.     

一个截断误差诱导下的随机数字振荡系统

计算机截断误差通常被认为会导致混沌系统退化.根据这种认识，提供了一个完全由计算机截断误差诱导的简单系统走向复杂化或混沌的反例.该系统定义为椭圆反射腔映射系统的过焦系统，理论解为一个极限序列，对应计算机解则是一个随机数字振荡系统.分析表明，计算机解是在截断误差诱导下由理论解变异而来的.理论解中系统存在“回转机制”，截断误差诱导系统在非双曲性不动点局域产生“逃逸机制”，从而发现一种阵发混沌的新机制. 关键词： 截断误差 切延迟椭圆反射腔映射系统 随机振荡 阵发混沌     

用直线稳定方法控制耦合哈密顿系统的混沌运动

讨论了不稳定不动点邻域的不稳定轨道的稳定问题．通过对系统施加外部的控制信号，将直线稳定方法推广到控制高维保守系统一耦合标准映象的混沌运动．通过对外加控制信号的调整，使系统不稳定不动点邻域的不稳定轨道沿着连接任意时刻轨道点和该不动点的直线趋向不动点，从而使难于控制的高维保守系统的不稳定轨道趋于稳定．这种方法不需要事先掌握系统动力行为，而且具有较强的抗干扰能力。     

截断误差导致的非双曲不动点邻域拓扑变异

提供一个关于截断误差使简单系统复杂化的直接实验证据,以此证明存在混沌抗退化机理.分别构造了一个一维圆弧迭代系统和一个一维抛物线迭代系统,两者均有一个非双曲不动点,其迭代序列被证明是简单极限序列,数字计算实验显示这两个迭代系统都存在可以越过不动点的序列.采用计算"元胞"分析方法清晰地展示了截断误差导致非双曲不动点邻域拓扑变异:形成第I类阵发混沌通道,或产生纹波分岔.     

预测反馈法控制双向闭环时空混沌系统的理论及实验

预测反馈控制方法可以用于控制时空混沌系统,该方法是在耦合映象格子中的每个格点处加入局部预测反馈控制器.本文以双向环形Henon耦合映象格子为例,在理论上给出了将系统控制到不稳定不动点的充分条件,并通过数值计算及电路仿真实验证实该方法的有效性.     

耦合双稳映象格子模型的时空混沌控制

变量反馈技术实现了耦合双稳映象格子模型的时空混沌控制.数值实验结果表明,利用不同的反馈技术和不同的反馈强度,可以将双稳映象系统的混沌及耦合双稳映象格子模型的时空混沌控制到不动点或周期轨道.变量反馈控制法除了局域双稳映象系统的定态点外,不需要先获取耦合双稳映象格子时空系统的动力学信息,它对抑制耦合双稳映象系统中的湍流具有一定的指导作用.     

基于自适应脉冲微扰实现混沌控制的研究

提出了自适应脉冲微扰控制混沌系统的方法.在参量脉冲微扰中引入自适应控制策略，设计出可以产生合适的脉冲强度的自适应控制器来实现混沌控制.采取这种方法对混沌的Rssle r连续系统和Hnon离散映射实施仿真控制，能够将系统稳定到不同的周期轨道或不动点上 ；并且，数值仿真结果还表明该控制方法具有较强的鲁棒性. 关键词： 自适应 脉冲微扰 混沌控制 鲁棒性     

New type of heteroclinic tangency in two-dimensional maps

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.     

A Prelude Staircase to a Type V Intermittency in Two-Dimensional Discontinuous Maps

A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which canbe considered as a model of impact oscillator. The so-called "transfer number", which is defined as the mean numberof transfer from non-impact state to impact state per iteration, is locked onto a lot of rational values to form a curveconsisting of many steps. Our numerical investigation confirms that every step is confined by conditions created by thecollision between the periodic orbit and the discontinuous boundary of the system. After the last collision the systemshows a chaotic motion with intermittent characteristics. Therefore the staircase can be addressed as a "prelude staircaseto type V intermittency". The similar phenomenon has also been observed in a model of electric circuit. These resultsof our study suggest that this kind of staircases is common in two (or even higher) dimensional discontinuous maps.     

Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models

We present a qualitative analysis of a generic model structure that can simulate the bursting and spiking dynamics of many biological cells. Four different scenarios for the emergence of bursting are described. In this connection a number of theorems are stated concerning the relation between the phase portraits of the fast subsystem and the global behavior of the full model. It is emphasized that the onset of bursting involves the formation of a homoclinic orbit that travels along the route of the bursting oscillations and, hence, cannot be explained in terms of bifurcations in the fast subsystem. In one of the scenarios, the bursting oscillations arise in a homoclinic bifurcation in which the one-dimensional (1D) stable manifold of a saddle point becomes attracting to its whole 2D unstable manifold. This type of homoclinic bifurcation, and the complex behavior that it can produce, have not previously been examined in detail. We derive a 2D flow-defined map for this situation and show how the map transforms a disk-shaped cross-section of the flow into an annulus. Preliminary investigations of the stable dynamics of this map show that it produces an interesting cascade of alternating pitchfork and boundary collision bifurcations. Received 24 June 1999 and Received in final form 17 February 2000     

Noise-induced intermittency in the quasiperiodic regime of Rayleigh-Bénard convection

Thermal measurements on a converting dilute³He-superfluid⁴He solution in the quasiperiodic regime show a transition from a mode-locked periodic state to chaotic time dependence via intermittency. The type of intermittency is discussed in the context of standard models of the phenomenon. In a region just below the onset of intermittency, injection of external multiplicative noise with noise amplitude above a certain threshold level induces the chaotic state. This noise-induced transition can be understood to be due to perturbations of a system with a barely stable attractor; the noise causes the system to escape the weakly attracting periodic points. We present a numerical simulation of a 1D map with external noise which explains some aspects of the noise-induced behavior, and a 2D map which has certain features of the intermittency.     

The timing of information transmission in financial markets

This article shows turbulent behavior in a series of financial indexes assuming that they follow a cascade process of the same type as do turbulent fluids. With such a model, the energy flux between the eddies that emerge in the fluid is analogous to the financial information flux over the course of time. The results obtained confirm the variability of variation of the indexes for the considered time scale (the turbulent intermittency typical for fluids), and they also confirm that when we descend along the cascade, that is to say, when we consider smaller time intervals, the rate at which the hypothetical eddies of information dissipate becomes greater than the rate at which the information is transmitted. This fact can explain the cyclical nature of crises: ultimately, financial events have a memory of the past. Besides, the NASDAQ singular behavior regarding the number of jumps, the degree of intermittency of the turbulence and the life time of the hypothetical eddies has been analysed.     

具有两个边界的Boost变换器分岔行为和斜坡补偿的镇定控制

电流控制型Boost变换器在较宽的电路参数下具有两个边界,建立了采用斜坡补偿电流的分段光滑迭代映射方程,并导出了轨道状态发生转移时的分界线方程,通过数值仿真得到了输入电压和斜坡补偿斜率变化时的逆分岔图和它们的动力学行为分布图.研究结果表明,随着输入电压逐步减小,Boost变换器从稳定的周期1态,经在边界1上发生边界碰撞分岔后进入连续传导模式（CCM）下的鲁棒混沌态,并经在边界2上发生边界碰撞分岔后进入不连续传导模式（DCM）下的强阵发性的弱混沌态.通过引入合适的斜坡补偿电流,Boost变换器的工作模式可以 关键词： Boost变换器 斜坡补偿 迭代映射方程 镇定控制     

Statistical analysis of the transition to turbulence in plane Couette flow

We argue on general grounds that the transition to turbulence in plane Couette flow is best studied experimentally at a statistical level. We present such a statistical analysis of experimental data guided by a parallel investigation of a simple coupled map lattice model for spatiotemporal intermittency. We confirm that this generic type of spatiotemporal chaos is relevant in the context of plane Couette flow, where the linear stability of the laminar regime at all Reynolds numbers insures the necessary local subcriticality. Using large ensembles of similar experiments, we show the existence of a well-defined threshold Reynolds number above which a unique, turbulent, intermittent attractor coexists with the laminar flow. Furthermore, our data reveals that this transition to spatiotemporal intermittency is discontinuous, i.e. akin to a first-order phase transition. Received: 10 April 1998 / Revised: 22 June 1998 / Accepted: 24 June 1998     

Renormalization for golden circles

We prove that twist maps of the cylinder that are attracted by any fixed point of MacKay's renormalization operator have a transitive invariant golden circle, provided the fixed point satisfies a few simple, purely topological conditions. These conditions can be verified by finite-precision arithmetics; they are fulfilled for the simple fixed point and seem to be fulfilled for the critical fixed point. Taking existence and hyperbolicity of the critical fixed point for granted, we conclude that the standard map has a critical invariant golden circle; the induced map on the circle is topologically conjugate to a rigid rotation; we can show that the conjugator is Hölder continuous; moreover, it is not differentiable on a dense set of points.This paper is part of a PhD thesis that is in preparation under the supervision of Oscar E. Lanford III at the ETH. I thank Oscar Lanford for having asked me precisely the right questions.     

Robust dangerous border-collision bifurcations in piecewise smooth systems

Physical and computer experiments involving systems describable by piecewise smooth continuous maps that are nondifferentiable on some surface in phase space exhibit novel types of bifurcations in which an attracting fixed point exists before and after the bifurcation. The striking feature of these bifurcations is that they typically lead to "unbounded behavior" of orbits as a system parameter is slowly varied through its bifurcation value. This new type of border-collision bifurcation is fundamental and robust. A method that prevents such "dangerous border-collision bifurcations" is given. These bifurcations may be found in a variety of experiments including circuits.