标准映象的周期分岔及其向混沌的过渡

本文研究了标准映象的周期轨道在其余数R=1和R=0附近的行为,对于前者出现倍周期分岔序列,其分岔率δ及标度因子α和β与其它二维保面积映象的结果一致;对于后者发生同周期分岔,这与标准映象的奇对称性有关。文中还计算了混沌轨道的李雅普诺夫指数,发现在倍周期分岔序列的聚点k_∞附近,存在标度关系: λ=λ_∞+A(k—k_∞)+B(k—k_∞)^τ, 其中因子τ=0.32。这与理论推断的结果τ=(ln(2))/(l 关键词：     

一个张弛振子的超临界子区域

解析导出了描述一种电子张弛振子的分段光滑圆映象在参数空间超临界区域内显示几种不同动力学行为的子区域分界线.这些子区域分别是：允许倍周期分岔发生的区域,禁止倍周期分岔发生的区域,以及完全锁相区域.类似的现象和子区域可以在许多分段光滑系统中观察到. 关键词：     

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     

二维均匀耦合映象格子中的时空周期图案

目的——构造二维均匀耦合映象格子中的时空周期图案;方法——通过一维耦合映象格子模型的相空间中已知低空间周期轨道,直接构造二维均匀耦合映象格子模型中一系列空间周期轨道,而不必求解其模型方程,并对构造轨道的稳定性进行分析;结果——L²×L²雅可比矩阵可化简为几个2×2矩阵组成的对角矩阵;结论——所构造轨道的稳定性不可能比原来轨道的稳定性高. 关键词： 耦合映象格子 时空周期图案 雅可比矩阵     

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     

一个强、弱耗散功能可分隔系统中的特征激变

报道一个由保守映象和耗散映象不连续、不可逆地分段描述的系统，以及在其中发生的一例特征激变.激变的独特之处在于逃逸孔洞.由映象的不连续、不可逆性而导致相平面中出现一个胖分形迭代禁区网，它使得一个混沌吸引子突然失稳而发生激变后出现的两个周期吸引子的吸引域边界成为点滴状.仅仅在每个周期点邻近存在这样的一个作为逃逸孔洞的、受到强耗散性支配和禁区边界限制的规则边界吸引域. 关键词： 激变 保守映象 耗散映象 逃逸孔洞     

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.     

小型短电磁脉冲传感器

对一种短偶极子类小型短电磁脉冲传感器进行了研究,分析了传感器电容、有效面积与结构参数之间的解析关系,给出了传感器的工作原理。根据应用需求设计了不同带宽的该类传感器,并对其波形保真性和有效面积进行了数值分析,仿真结果表明:经传感器接收并还原所得场波形与激励电场波形几乎完全重合,传感器有效面积与理论分析结果偏差小于1%。对传感器的焊接、定位和对轴等制作工艺进行了研究,完成了传感器样品的制作。利用单锥TEM室对传感器的波形保真性和有效面积进行了实验测试,测试结果表明:设计制作的传感器样品可以很好地恢复待测脉冲电场的波形和幅度,两支传感器样品的有效面积实测结果与解析计算结果较为一致,偏差分别小于4%和7%。     

不可微不可逆映象中的V型阵发

一个既不可微(或不连续)又不可逆的一维映象可以展示一种新型的阵发。它的机制是稳定不动点与映象不可微或不连续点碰撞而消失。这种阵发可以在该不动点附近的线性化映象本征值绝对值在阵发前为小于1的任何值的情况发生,因而可能突然出现在倍周期分岔序列中途任一部分,使序列中断进入混沌。在稳定不动点消失后映象产生的阵发时间序列中,层流相长度呈现与外控参数距临界值距离的对数依赖关系。这种新型标度规律不依赖于映象的细节。作者认为这种阵发应广泛存在于许多实际系统之中。 关键词：     

周期扰动控制一维映象混沌运动

从一维映象出发提出了周期扰动控制混沌的方法，给出了针对一动力学目标和估计扰动幅度的方法，以Ｌｏｇｉｓｉｃ映象为例数值夺研究了这种控制方法的各种性质。     

A linear code for the sawtooth and cat maps

The sawtooth maps are a one-parameter set of piecewise linear area preserving maps on the torus. For positive integer values of the parameter K they are automorphisms of the torus, known as the cat maps. We present a symbolic dynamics for these maps in which the symbols are integers. This code is related to a practical problem of the stabilisation of a system which is perturbed by impulses. The code is linear in the sense that an orbit and its code are linearly related, so it is not difficult to obtain a good approximation to one from the other in practice. A stationary stochastic process for generating the code is given explicitly. The theory uses Green function methods, which are also used to study ordered periodic orbits and cantori. The problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed.     

The Plateau and Symmetrical Structure of Lyapunov Exponents in 2-Dimensional Homogeneous. Dissipative Map

The universal crossover behavior of Lyapunov exponents in transition from conservative limit to dissipative limit of dynamical system is studied. We discover numerically and prove analytically that for homogeneous dissipative two-dimensional maps, along the equal dissipation line in parameter space, two Lyapunov exponents λ₁ and λ₂ of periodic orbits possess a plateau structure, and around this exponent plateau value, there is a strict symmetrical relation between λ₁ and λ₂ no matter whether the orbit is periodic, quasiperiodic, or chaotic.The method calculating stable window and Lyapunov exponent plateau widths is given. For Hénon map and 2-dimensional circle map, the analytical and numerical results of plateau structure of Lyapunov exponents for period-1,2 and 3 orbits are presented.     

The periodic orbits of an area preserving twist map

We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.     

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.     

Periodic attractors of random truncator maps

This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a non-commutative algebra and classify its periodic orbits using properties of endomorphisms of the resulting algebraic structure. A stochastic model is constructed on these endomorphisms, which leads to the classification of the distribution of periodic orbits for random truncator maps. This framework is applied to investigate the periodic transitions of Bornholdt's spin market model.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

Some flesh on the skeleton: The bifurcation structure of bimodal maps

One of the main tools in the numerical study of two-parameter families of one-dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits. We use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points. Then we explain how the bifurcation structure hangs on this “skeleton”.     

Statistical properties of actions of periodic orbits

We investigate statistical properties of unstable periodic orbits, especially actions for two simple linear maps (p-adic Baker map and sawtooth map). The action of periodic orbits for both maps is written in terms of symbolic dynamics. As a result, the expression of action for both maps becomes a Hamiltonian of one-dimensional spin systems with the exponential-type pair interaction. Numerical work is done for enumerating periodic orbits. It is shown that after symmetry reduction, the dyadic Baker map is close to generic systems, and the p-adic Baker map and sawtooth map with noninteger K are also close to generic systems. For the dyadic Baker map, the trace of the quantum time-evolution operator is semiclassically evaluated by employing the method of Phys. Rev. E 49, R963 (1994). Finally, using the result of this and with a mathematical tool, it is shown that, indeed, the actions of the periodic orbits for the dyadic Baker map with symmetry reduction obey the uniform distribution modulo 1 asymptotically as the period goes to infinity. (c) 2000 American Institute of Physics.     

Distribution of periodic orbits for the Casati-Prosen map on rational lattices

We study numerically the periodic orbits of the Casati-Prosen map, a two-parameter reversible map of the torus, with zero entropy. For rational parameter values, this map preserves rational lattices, and each lattice decomposes into periodic orbits. We consider the distribution function of the periods over prime lattices, and its dependence on the parameters of the map. Based on extensive numerical evidence, we conjecture that, asymptotically, almost all orbits are symmetric, and that for a set of rational parameters having full density, the distribution function approaches the gamma-distribution R(x)=1−e^−x(1+x). These properties, which have been proved to hold for random reversible maps, were previously thought to require a stronger form of deterministic randomness, such as that displayed by rational automorphisms over finite fields. Furthermore, we show that the gamma-distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space. Our experiments reveal that the convergence rate to R is highly non-uniform in parameter space, being slowest in sharply-defined regions reminiscent of resonant zones in Hamiltonian perturbation theory.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.