Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.     

Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.     

Recurrence network analysis of experimental signals from bubbly oil-in-water flows

Based on the signals from oil–water two-phase flow experiment, we construct and analyze recurrence networks to characterize the dynamic behavior of different flow patterns. We first take a chaotic time series as an example to demonstrate that the local property of recurrence network allows characterizing chaotic dynamics. Then we construct recurrence networks for different oil-in-water flow patterns and investigate the local property of each constructed network, respectively. The results indicate that the local topological statistic of recurrence network is very sensitive to the transitions of flow patterns and allows uncovering the dynamic flow behavior associated with chaotic unstable periodic orbits.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

Polar nano-rods under shear: From equilibrium to chaos

The orientational dynamics of rod-like particles with permanent (electric or magnetic) dipole moments in a plane Couette shear flow is investigated using mesoscopic relaxation equations combined with a generalized Landau free energy. The free energy contribution due to the coupling between average alignment and dipole orientation is derived on a microscopic basis. Numerical results of the resulting eight-dimensional dynamical system are presented for the case of longitudinal dipoles and thermodynamic conditions where the equilibrium state is a (polar or non-polar) nematic. Solution diagrams reveal presence of a large variety of periodic, transient chaotic, and chaotic dynamic states of the average alignment and dipole moment, respectively, appearing as a function of Deborah number and tumbling parameter. Compared to rods without dipoles we observe a significant preference of out-of-plane kayaking-tumbling states and, generally, a higher sensitivity to the initial conditions including bistability. We also demonstrate that the average (electric) dipole moment characterizing most of the observed states yields electrodynamic (magnetic) fields of measurable strength.     

基于混沌吸引子形态特性的两相流流型分类方法研究

利用高灵敏度差压传感器，在垂直上升管中采集到了80组气液两相流差压波动信号.利用放置参考截面的方法，建立了描述混沌吸引子形态的一般方法，在此基础上，提出了不同维数下的吸引子形态特征量进行组合的气液两相流流型分类新方法.研究结果表明：该方法对包括复杂过渡流型在内的气液两相流流型有很好分类效果，预示着混沌吸引子形态描述是研究非线性时间序列的实用有效途径. 关键词： 气液两相流 流型分类 吸引子形态 混合维     

Can renormalization group flow end in a Big Mess?

The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan–Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.     

Dynamics of zero-Prandtl number convection near onset

We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Be?nard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the most energetic modes of DNS. The bifurcation analysis reveals a rich variety of convective flow patterns and chaotic solutions, some of which are common to that of the 13-mode model of Pal et al. [EPL 87, 54003 (2009)]. We also observed a set of periodic and chaotic wavy rolls in DNS and in the model similar to those observed in experiments and numerical simulations. The time period of the wavy rolls is closely related to the eigenvalues of the stability matrix of the Hopf bifurcation points at the onset of convection. This time period is in good agreement with the experimental results for low-Prandtl number fluids. The chaotic attractor of the wavy roll solutions is born through a quasiperiodic and phase-locking route to chaos.     

Hybrid approach to high-frequency microfluidic mixing

We report the experimental verification of the predicted chaotic mixing characteristics for a polydimethylsioxane microfluidic chip, based on the mechanism of multistage cross-channel flows. While chaotic mixing can be achieved within short passage distances, there is an optimal side channel flow pulsation frequency beyond which the mixing becomes ineffective. Based on the physical understanding of a Poincaré section analysis, we propose the installation of passive flow baffles in the main microfluidic channel to facilitate high-frequency mixing. The combined hybrid approach enables chaotic mixing at enhanced frequency and reduced passage distance in two-dimensional flows.     

混沌实验数据处理及仿真

大学物理实验中混沌实验大多采用观察现象的方法进行,本实验采用蔡氏电路(Chua s circuit)产生混沌行为.在观察不同初始值条件下出现的倍周期分岔、阵发混沌、奇异吸引子等相图及现象的基础上,通过对采集数据进行处理,对负电阻伏安特性进行分段线性拟合,用功率频谱法、计算机仿真方法(龙格-库塔数值积分法)对混沌现象进行描绘,将实验数据与非线性方程组的数值解相结合,呈现出混沌现象的本质.     

Switching phenomenon induced by breakdown of chaotic phase synchronization

We investigate chaotic phase synchronization (CPS) in three-coupled chaotic oscillator systems. According to the coupling strength and mismatches in the frequencies of these oscillators, we can observe complete CPS where all three oscillators exhibit CPS, and partial CPS where only two oscillators exhibit CPS. When the coupling strength is weakened, we observe a phenomenon that complete CPS among the three oscillators is suddenly disrupted without going through partial CPS. In this case oscillators exhibit quasi-CPS where two oscillators appear to exhibit CPS transiently, and the combination of the two oscillators changes with time. We call this phenomenon CPS switching D. It is revealed that phase fluctuation plays an important role in CPS switching D. It is also shown that the amplitude with a specific structure strengthens the degree of CPS switching. In the present paper, we characterize this CPS switching and discuss its mechanism.     

Chaotic dynamics of erbium-doped fiber laser with nonlinear optical loop mirror

We experimentally and numerically demonstrate the chaotic dynamics of the erbium-doped fiber laser with a nonlinear optical loop mirror. When the polarization controllers are fixed at an appropriate orientation, we observe that the fiber laser exhibits a period-doubling route to chaos with increasing the pump power in the experiment and simulation. The numerical simulation shows a good agreement with the experimental results. The results show experimentally and numerically that the chaotic dynamics of the erbium-doped fiber laser is related to the polarization state and the pump power of light in the cavity.     

Chen氏混沌电路实现与同步控制实验研究

对Chen氏混沌系统的电路实现作了进一步的研究，推导出切实可行的电路参数.在此基础上利用Electronic Workbench电路仿真软件进行了仿真实验，并搭建实现了实际的Chen氏电路，在示波器上观察到了Chen氏混沌系统的混沌波形.同时，利用单变量耦合反馈控制方法对两个Chen氏系统的同步控制进行了实验研究，给出了实现同步的电路参数调节范围. 关键词： 混沌系统 Chen氏混沌 同步控制 电路实现     

两相流相空间多元图重心轨迹动力学特征

本文提出了一种新的混沌时间序列高维相空间多元图重心轨迹动力学特征提取方法. 在确定了最佳嵌入维数和延迟时间后, 将相空间中高维矢量点映射到二维平面的雷达图上, 相应地将相空间中高维矢量点变换为对应的几何多边形. 通过提取几何多边形的重心位置得到重心轨迹动力学演化特性, 并利用重心轨迹矩特征量区分不同性质的混沌时间序列. 在此基础上, 处理分析了气液两相流电导传感器动态信号, 发现高维相空间多元图重心轨迹矩特征量不仅可以辨识泡状流、段塞流和混状流, 而且为流型动力学演化机理提供了新的分析途径.     

Cycles homoclinic to chaotic sets; robustness and resonance

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient 'bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called 'cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821-823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rossler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. (c) 1997 American Institute of Physics.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Coevolutionary motion and swarming in a niche space model of ecological species interactions

Organisms are involved in coevolutionary relationships with their competitors, predators, preys and parasites. In this context, we present a simple model for the co-evolution of species in a common niche space, where the fitness of each species is defined via the network of interactions with all other species. In our model, the sign and type of the pairwise interactions (being either beneficial, harmful or neutral) is given by a pre-determined community matrix, while the interaction strength depends on the niche-overlap, i.e. the pairwise distances between species in niche space. The evolutionary process drives the species toward the places with the higher local fitness along the fitness gradient. This gives rise to a dynamic fitness landscape, since the evolutionary motion of a single species can change the landscape of the others (known as the Red Queen Principle). In the simplest case of only two-species we observe either a convergence/divergence equilibrium or a coevolutionary arms race. For a larger number of species our analysis concentrates on an antisymmetric interaction matrix, where we observe a large range of dynamic behaviour, from oscillations, quasiperiodic to chaotic dynamics. In dependence of the value of a first integral of motion we observe either quasiperiodic motion around a central region in niche space or unbounded movement, characterised by chaotic scattering of species pairs. Finally, in a linear food-chain we observe complex swarming behaviour in which the swarm moves as a whole only if the chain consists of an even number of species. Our results could be an important contribution to evolutionary niche theory.     

摇摆条件下自然循环系统流量混沌脉动的检验与预测

利用替代数据法检验了摇摆条件下自然循环系统不规则复合型脉动的混沌特性, 并在此基础上进行混沌预测. 关联维数、最大Lyapunov指数等几何不变量计算结果表明不规则复合型脉动具有混沌特性, 但是由于计算结果受实验时间序列长度的限制和噪声的影响, 可能会出现错误的判断结果. 为了避免出现误判, 在提取流量脉动的非线性特征的同时, 需要用替代数据法进一步检验混沌特性是否来自于确定性的非线性系统. 本文用迭代的幅度调节Fourier 算法进行混沌检验, 在此基础上用加权一阶局域法进行混沌脉动的预测. 计算结果表明: 不规则复合型脉动是来自于确定性系统的混沌脉动, 加权一阶局域法对流量脉动进行混沌预测效果较好, 并提出动态预测方法. 关键词： 混沌时间序列 替代数据法 实时预测 两相流动不稳定性     

基于Lyapunov指数的摇摆条件下自然循环流动不稳定性混沌预测

运用基于最大Lyapunov指数的混沌预测方法对摇摆条件下自然循环系统的流量脉动进行了预测. 对不规则复合型脉动的流量脉动实验数据进行相空间重构, 计算关联维数、二阶Kolmogorov熵和最大Lyapunov指数等几何不变量, 在说明不规则复合型脉动是混沌运动的基础上, 根据最大Lyapunov指数对不规则复合型脉动进行了预测. 通过预测结果和实验结果对比发现: 对于复杂的两相自然循环流动不稳定性, 预测结果具有较高的精度, 说明预测方法的可行性. 同时, 确定了混沌系统可预测的尺度, 提出用动态预测的方式监测系统流量脉动. 本文的研究方法为两相流复杂的流动不稳定性研究提供了新的思路. 关键词： 混沌时间序列 实时预测 最大Lyapunov指数 两相流动不稳定性     

Transition to chaos in lid-driven square cavity flow

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid-driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.