周期受击陀螺系统波函数的分形

研究了周期受击陀螺系统波函数的分形. 发现在打击强度系数较弱时 (即≤ 1时), 相空间是规则的, 分形维接近于1; 随着打击强度系数的增大, 相空间开始变得混沌, 分形维也随之增大; 当打击强度系数达到6时, 相空间完全混沌, 分形维将达到最大值, 此时若继续增大打击强度系数, 分形维保持基本不变. 关键词： 陀螺 波函数 分形维 相空间     

Extensive chaos in the Lorenz-96 model

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.     

竖壁液膜流壁面热流率对流动影响的实验研究

本文用相空间重构的方法,对竖壁薄液膜流动的液膜厚度时间序列进行了重构,并用分维数对奇怪吸引子的动力特征进行了描述,由此描述了壁面热流率对竖壁薄液膜流动特性的影响。     

分形理论在光谱识别中的应用

分形理论是研究一类不规则、混乱复杂,但其局部和整体具有相似性体系的科学。分形维数是分形理论中用于描述对象的不规则度和自相似性的基本度量。文章以符合朗伯-比尔定律的光谱信号为研究对象,在概述分形几何基本原理的基础上,提出了以分形维数作为光谱识别特征的方法,运用相空间重构得出了光谱信号的分形维数,通过对光谱信号的分形维数进行比较,达到识别不同光谱的目的,最后举例对该方法进行了说明。     

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

Fractal Aggregation Under Rotation

By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω; thefractal dimension decreases with increasing ω;, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.     

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.     

Hyperchaos evolved from the Liu chaotic system

This paper introduces a new hyperchaotic system by adding an additional state into the third-order Liu chaotic system. Some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponent, fractal dimension and the hyperchaotic attractor evolving into chaotic, periodic, quasi-periodic dynamical behaviours by varying parameter d are studied briefly. Various attractors are illustrated not only by computer simulation but also by conducting an electronic circuit experiment.     

三维单向拉伸马蹄的寻找及其在Compass行走模型中的应用

对三维映射,因维数较高,混沌吸引子结构复杂.根据混沌吸引子维数相对较低的特点,在选定不稳定周期轨的附近,对吸引子的局部进行曲面拟合,提出了三维空间中单向拉伸拓扑马蹄的“降维-升维”寻找方法,并实现为MATLAB工具箱.以Compass行走模型为例,详细介绍拓扑马蹄的寻找过程,判定了混沌步态的存在性,刻画了混沌不变集的部分结构.     

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

Multiple coexisting attractors,Basin boundaries and basic sets

Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique which restricts orbits to the boundary. We call these numerically obtained orbits “straddle orbits”. By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum which has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high-dimensional system (five phase space dimensions), namely, two coupled driven Van der Pol oscillators. Two parameter sets are examined for this system. In one of these cases the basin boundaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basic sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basic sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the other quasiperiodic.     

Unexpected scenario of glass transition in polymer globules: an exactly enumerable model

We introduce a lattice model of glass transition in polymer globules. This model exhibits ergodicity breaking in which the disjoint regions of phase space do not arise uniformly, but as small chambers whose number increases exponentially with polymer density. Chamber sizes obey power law distribution, making phase space similar to a fractal foam. This clearly demonstrates the importance of the phase space geometry and topology in describing any glass-forming system, such as semicompact polymers during protein folding.     

研究多中心奇异吸引子混沌相位的新方法

对一般混沌轨迹的相位定义进行了研究,考虑到大量的混沌运动投影到相平面后不是只有一个转动中心的正规转动,给出了两种计算瞬时相位的简单方法.一种方法称之为直接分解方法,它是基于分解的思想.另一种是切线方法,它需要计算轨迹在切空间中的角度.通过对一些算例结果的比较,可以看出这两种方法的优越性.综合而言,切线方法是对任意混沌振动适用的,因此可以被广泛应用于混沌相位同步的研究中 关键词： 混沌同步 相位同步 正规转动     

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.     

一种混沌海杂波背景下的微弱信号检测方法

基于复杂非线性系统的相空间重构理论,提出了一种基于遗传算法的支持向量机预测方法.利用改进的自相关法和饱和关联维数法确定混沌信号的时间延迟和嵌入维,从而实现相空间重构.通过遗传算法优化支持向量机中的惩罚系数和核函数参数,并结合支持向量机建立混沌序列的单步预测模型,从预测误差中检测出淹没在混沌背景中的微弱信号(包括瞬态信号和周期信号).以Lorenz系统和加拿大McMaster大学利用IPIX雷达实测得到的海杂波数据作为混沌背景噪声进行仿真实验,结果表明该方法能够有效地从混沌背景噪声中检测出微弱目标信号,所得的均方根误差为0.00049521(信噪比为-89.7704 dB),这比传统支持向量机方法的均方根误差(0.049,信噪比为-54.60 dB)降低了两个数量级.     

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).     

Spatiotemporal dynamics in the coherence collapsed regime of semiconductor lasers with optical feedback

This paper presents a spatiotemporal characterization of the dynamics of a single-mode semiconductor laser with optical feedback. I use the two-dimensional representation of a time-delayed system (where the delay time plays the role of a space variable) to represent the time evolution of the output intensity and the phase delay in the external cavity. For low feedback levels the laser output is generally periodic or quasiperiodic and with the 2D representation I obtain quasiperiodic patterns. For higher feedback levels the coherence collapsed regime arises, and in the 2D patterns the quasiperiodic structures break and "defects" appear. In this regime the patterns present features that resemble those of an extended spatiotemporally chaotic system. The 2D representation allows the recognition of two distinct types of transition to coherence collapse. As the feedback intensity grows the number of defects increases and the patterns become increasingly chaotic. As the delay time increases the number of defects in the patterns do not increase and there is a signature of the previous quasiperiodic structure that remains. The nature of the two transitions is understood by examining the behavior of various chaotic indicators (the field autocorrelation function, the Lyapunov spectrum, the fractal dimension, and the metric entropy) when the feedback intensity and the delay time vary. (c) 1997 American Institute of Physics.     

Special electronic structures of inverse spinels LiMVO<Subscript>4</Subscript>(M = Ni and Cu): a first-principles study

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d₀/2 where d₀ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.     

Floquet states and geometrical phase of a multi-level system in cyclic evolution

We study the electronic states of a mesoscopic system whose Hamiltonian has a complicated static multi-level energy structure and undergoes periodic evolution in time. By using the Floquet theory, we derive the quasienergies, the Floquet states, and the geometrical phase. It is shown numerically that the geometrical phase is strongly dependent on the evolution circuits in the parameter space and on the evolution frequency of the varying Hamiltonian. In some cases the nonadiabatic geometric phases can exhibit chaotic behavior. We also show a trend of phase compensation in pairs of states which could restore the phase coherence if the pairing occurs.     

Topology of trajectories of the 2D Navier-Stokes equations

In spectral form the 2D incompressible Navier-Stokes equations in a square periodic region will be represented by 430 complex Fourier amplitudes which correspond to isotropic truncation of the upper wave number 16. For small viscosity, we have found five equilibrium states I-V in the entire range of forcing; I-fixed point, II-circle, III-closed orbit, IV-torus, and V-chaos. The fixed-point equilibrium state is the laminar flow. As the forcing passes through a critical value, the fixed point evolves directly to equilibrium state III under a typical multimode forcing. The chaotic transition takes place on a 2-torus-like manifold (equilibrium state IV) which is the product space of a circle and the closed orbit of equilibrium state III, similar to the quasiperiodic 2-torus of Ruelle and Takens. For sufficiently large forcing, the evolution of equilibrium state V is nothing but a simulation of quasistationary 2D turbulence. From the Lyapunov exponents of turbulent flows, we have evaluated the constants in the theoretical results of Foias and his colleagues, which relate the determining mode and fractal dimension with the enstrophy dissipation wave number of 2D turbulence.