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     

一例从保守向类耗散的过渡

建议研究一类特殊的分段连续力场中的受击转子.它在一个控制参数连续改变时可以演示从分段连续保守系统向类耗散系统的连续过渡，从而展现不连续边界像集随机网向一个瞬态随机网，以及网上的无边界混沌扩散运动向局域规则运动的转变.这种转变可能用瞬态随机网的分数维随控制参数的对数改变规律来描述，也可以用迭代在瞬态随机网中的平均生存时间随控制参数的指数改变规律来描述. 关键词： 受击转子 分段连续保守系统 类耗散系统 随机网     

一例系统性质改变导致的激变

当控制参数改变时，在一个受击台球模型中观察到从处处光滑保守系统向分段光滑类耗散系统的过渡。它导致标志典型保守随机网向系统函数不连续边界象集构成的瞬态随机网突然转变的特殊激变。瞬态随机网上的迭代最终落入一个由椭圆岛链形成的，在上述转变阈值出现的逃逸孔洞。这孔洞随控制参数增长而变大，使迭代逃逸更快，因此瞬态网上迭代的平均生存时间遵从具有特殊标度因子的幂律。与此同时，一个在同一阈值出现的肥分形禁区网也不断增长而且切掉原来保守随机网的越来越多的部分，使得剩余的瞬态网越来越“瘦”。我们的数值研究表示这一过程可以用另一个幂律来描述。     

一个分段连续系统中的胖奇异集激变

报道一种有特色的激变.这种激变是在一类分段连续力场作用下的受击转子模型中观察到的.描述系统的二维映象定义域中的函数不连续边界随离散时间发展振荡，从而使这个边界的向前象集构成一个承载混沌运动的胖分形.在控制参数的一个阈值下，一个椭圆周期轨道突然出现在此胖混沌奇异集中，使得迭代向它逃逸，胖混沌奇异集因此突然变为一个胖瞬态集.在这种情况下，有可能根据椭圆周期轨道逃逸孔洞，以及胖分形奇异集的测度随参数变化的规律，估算迭代在奇异集中的平均生存时间所遵循的标度规律.直接数值计算和由此估算所得标度因子值可以很好地互相印证. 关键词： 激变 胖分形 分段连续系统 标度律     

一类不可逆保守系统中的混沌类吸引子

报道一类不可逆保守系统的性质及在其中发现的混沌类吸引子.这些系统可以看作是一种加过压保护的张弛振荡电路的简化模型.它展示的类吸引性是指这个混沌轨道吸引它之外的迭代,而这种吸引由系统的不可逆性所导致.数值研究发现这个混沌轨道就是系统不连续边界的映象集,而且这很可能是这类系统的普遍性质. 关键词： 不可逆保守系统 混沌类吸引子 张弛振荡     

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     

一类新的边界激变现象:混沌的边界激变

混沌吸引子的激变是一类普遍现象.借助于广义胞映射图论(generalized cell mapping digraph)方法发现了嵌入在分形吸引域边界内的混沌鞍,这个混沌鞍由于碰撞混沌吸引子导致混沌吸引子完全突然消失,是一类新的边界激变现象,称为混沌的边界激变.可以证明混沌的边界激变是由于混沌吸引子与分形吸引域边界上的混沌鞍相碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混沌吸引子连同它的吸引域突然消失,同时这个混沌鞍也突然增大 关键词： 广义胞映射 有向图 激变 混沌鞍     

一个二维分段光滑映象中的边界激变

解析地讨论一个冲击振子模型中的边界激变特征,证明了这类分段光滑二维映象中激变的生 存时间标度律为τ-ε^-γ,而γ=ln|β₂|ln|β₁β2|(β₁,β₂分别是混沌吸引子吸引域边界上鞍点的不稳 定和稳定本征值),这与处处光滑二维映象中相应的规律完全不同. 关键词： 激变 分段光滑映象 生存时间标度律     

常微分方程系统中内部激变现象的研究

应用广义胞映射图论方法研究常微分方程系统的激变.揭示了边界激变是由于混沌吸引子与 在其吸引域边界上的周期鞍碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混 沌吸引子连同它的吸引域突然消失,在相空间原混沌吸引子的位置上留下了一个混沌鞍.研 究混沌吸引子大小(尺寸和形状)的突然变化,即内部激变.发现这种混沌吸引子大小的突然 变化是由于混沌吸引子与在其吸引域内部的混沌鞍碰撞产生的,这个混沌鞍是相空间非吸引 的不变集,代表内部激变后混沌吸引子新增的一部分.同时研究了这个混沌鞍的形成与演化. 给出了对永久自循环胞集和瞬态自循环胞集进行局部细化的方法. 关键词： 广义胞映射 有向图 激变 混沌鞍     

一类含非黏滞阻尼的Duffing单边碰撞系统的激变研究

以一类含非黏滞阻尼的Duffing单边碰撞系统为研究对象, 运用复合胞坐标系方法, 分析了该系统的全局分岔特性. 对于非黏滞阻尼模型而言, 它与物体运动速度的时间历程相关, 能更真实地反映出结构材料的能量耗散现象. 研究发现, 随着阻尼系数、松弛参数及恢复系数的变化, 系统发生两类激变现象: 一种是混沌吸引子与其吸引域内的混沌鞍发生碰撞而产生的内部激变, 另一种是混沌吸引子与吸引域边界上的周期鞍(混沌鞍)发生碰撞而产生的常规边界激变(混沌边界激变), 这两类激变都使得混沌吸引子的形状发生突然改变. 关键词： 非黏滞阻尼 Duffing碰撞振动系统 激变 复合胞坐标系方法     

The properties of borderlines in discontinuous conservative systems

The properties of the set of borderline images in discontinuous conservative systems are commonly investigated. The invertible system in which a stochastic web was found in 1999 is re-discussed here. The result shows that the set of images of the borderline actually forms the same stochastic web. The web has two typical local fine structures. Firstly, in some parts of the web the borderline crosses the manifold of hyperbolic points so that the chaotic diffusion is damped greatly; secondly, in other parts of phase space many holes and elliptic islands appear in the stochastic layer. This local structure shows infinite self-similarity. The noninvertible system in which the so-called chaotic quasi-attractor was found in [X.-M. Wang et al., Eur. Phys. J. D 19, 119 (2002)] is also studied here. The numerical investigation shows that such a chaotic quasi-attractor is confined by the preceding lower order images of the borderline. The mechanism of this confinement is revealed: a forbidden zone exists that any orbit can not visit, which is the sub-phase space of one side of the first image of the borderline. Each order of the images of the forbidden zone can be qualitatively divided into two sub-phase regions: one is the so-called escaping region that provides the orbit with an escaping channel, the other is the so-called dissipative region where the contraction of phase space occurs.     

Exact ground state of a frustrated integer-spin modified Shastry-Sutherland model

We propose a method to measure real-valued time series irreversibility which combines two different tools: the horizontal visibility algorithm and the Kullback-Leibler divergence. This method maps a time series to a directed network according to a geometric criterion. The degree of irreversibility of the series is then estimated by the Kullback-Leibler divergence (i.e. the distinguishability) between the inand outdegree distributions of the associated graph. The method is computationally efficient and does not require any ad hoc symbolization process. We find that the method correctly distinguishes between reversible and irreversible stationary time series, including analytical and numerical studies of its performance for: (i) reversible stochastic processes (uncorrelated and Gaussian linearly correlated), (ii) irreversible stochastic processes (a discrete flashing ratchet in an asymmetric potential), (iii) reversible (conservative) and irreversible (dissipative) chaotic maps, and (iv) dissipative chaotic maps in the presence of noise. Two alternative graph functionals, the degree and the degree-degree distributions, can be used as the Kullback-Leibler divergence argument. The former is simpler and more intuitive and can be used as a benchmark, but in the case of an irreversible process with null net current, the degree-degree distribution has to be considered to identify the irreversible nature of the series.     

A semi-dissipative crisis

This article reports a sudden chaotic attractor change in a system described by a conservative and dissipative map concatenation. When the driving parameter passes a critical value, the chaotic attractor suddenly loses stability and turns into a transient chaotic web. The iterations spend super-long random jumps in the web, finally falling into several special escaping holes. Once in the holes, they are attracted monotonically to several periodic points. Following Grebogi, Ott, and Yorke, we address such a chaotic attractor change as a crisis. We numerically demonstrate that phase space areas occupied by the web and its complementary set (a fat fractal forbidden net) become the periodic points' “riddled-like” attraction basins. The basin areas are dominated by weaker dissipation called “quasi-dissipation”. Small areas, serving as special escape holes, are dominated by classical dissipation and bound by the forbidden region, but only in each periodic point's vicinity. Thus the crisis shows an escape from a riddled-like attraction basin. This feature influences the approximation of the scaling behavior of the crisis's averaged lifetime, which is analytically and numerically determined as 〈τ〉∝(b-b₀)^γ, where b₀ denotes the control parameter's critical threshold, and γ≃-1.5.     

Long time behavior in the quantized standard map with dissipation

A dissipative version of the quantized standard map is constructed by analytical means and iterated numerically to study the long time behavior in various regions of the damping rate. For weak dissipation, stochastic transitions induced by the heat bath disrupt the localization in the action variable, which suppresses chaotic motion in the conservative quantized standard map, and tend to restore diffusion of action. A steady state is reached on the time scale of classical relaxation. For strong dissipation, observable deviations from classical behavior both in the transients and in the statey state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.     

Periodically kicked hard oscillators

A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval; the topological structure of this object is generated by the finite size of the repelling set, and is therefore also intrinsic to hard oscillators.     

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.     

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.