Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.     

频率扫描法研究倍周期分岔与混沌现象

在一类非线性系统中,应用频率控制方法,对倍周期分岔与混沌行为进行了研究。在V₀-ω外控参数平面上,频率扫描显示了分岔与混沌的整体结构:正的和逆的倍周期分岔序列的对称性;分岔收敛于一点的封闭性。本文中所建议的方法,将是一种研究分岔与混沌现象有效而快速的手段。它不仅能定量测量收敛比δ和标度因子α,分段展开还能定性地观察阵发混沌和嵌套在混沌带中的各种窗口等。分岔与混沌是一类非线性系统的频率响应。 关键词：     

Characterization of chaos

We give a brief introduction to chaos and its characterization. We examine some standard systems in detail from the perspective of chaos and review their properties. Concepts necessary to understand them, such as dimension, are also reviewed. To illustrate the main ideas, we choose three examples which have served as paradigms for the study of chaos in physical systems, namely, the Hénon discrete mapping, the Lorenz system of coupled ordinary differential equations, and the Mackey-Glass infinite-dimensional delay differential equation.     

THE SWING VIBRATION, TRANSVERSE OSCILLATION OF CRACKED ROTOR AND THE INTERMITTENCE CHAOS

This paper studies the non-linear dynamic response of a cracked rotor by taking the swing vibration of disc into consideration. The results show that if a small crack appears, the frequency of transverse oscillation is synchronous with rotating speed ratio (Ω), and the frequency of swing vibration is N Ω (N=1,2,…). As the crack increases, the response becomes chaotic in some range of Ω. The deeper the crack is, the wider the chaotic range of Ω is. Routes to chaos include intermittence to chaos and quasi-period to chaos. When the crack is fairly deep, some new resonance regions develop. In these regions, the response becomes infinity rapidly. The appearance of intermittence chaos is induced by the frequent frustration of stable oscillation, which is resulted from the continuous increase of swing amplitude. Unbalance parameter U is effective in suppressing chaos. Crack angle β cannot affect the essence of response, but can influence the amplitude of synchronous response.     

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.     

A Class of Quantum System with Random Perturbation

In the presence of colored Gaussian noise, the quantum correspondence of Power system and its chaos anti-control are investigated. Some properties about the quantum chaos and classical chaos of the system are analyzed by adding the colored Gaussian noise to the phase of the system. Firstly, the quantization method is used to analyze some properties about the quantum correspondence of the classical chaos Power system. Then, the macroscopic dynamic behavior of the perturbed Power system is investigated. In addition, by using the computer simulation, we plot the Poincaré map and phase portraits to detect whether the system is chaos or not.     

Transient chaos in a generalized Hénon map on the torus

A generalized Hénon system on a torus is considered to investigate some phenomena of transient chaos in weakly dissipative systems. A simple relationship is numerically verified between the life time of chaos and other parameters of the model.     

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The dynamics of cold atoms in conservative optical lattices obviously depends on the geometry of the lattice. But very similar lattices may lead to deeply different dynamics. In a 2D optical lattice with a square mesh, it is expected that the coupling between the degrees of freedom leads to chaotic motions. However, in some conditions, chaos remains marginal. The aim of this paper is to understand the dynamical mechanisms inhibiting the appearance of chaos in such a case. As the quantum dynamics of a system is defined as a function of its classical dynamics – e.g. quantum chaos is defined as the quantum regime of a system whose classical dynamics is chaotic – we focus here on the dynamical regimes of classical atoms inside a well. We show that when chaos is inhibited, the motions in the two directions of space are frequency locked in most of the phase space, for most of the parameters of the lattice and atoms. This synchronization, not as strict as that of a dissipative system, is nevertheless a mechanism powerful enough to explain that chaos cannot appear in such conditions.     

Long way from the FPU-problem to chaos

This paper provides some historical comments on the study of the Fermi, Pasta, and Ulam (FPU) paper and its influence on the development of the theory of chaos. We also discuss some problems raised in the FPU paper and the links of these problems to such contemporary notions in chaos theory as ergodicity, mixing, recurrences, pseudochaos, kinetics, intermittency, etc.     

The nature of bursting noises,stochastic resonance and deterministic chaos in excitable neurons

We study the Hindmarsh-Rose model of excitable neurons and show that in the asymptotic limit this monostable model can possess some kind of dynamical bistability: small-amplitude quasiharmonic and large-amplitude relaxational oscillations can be simultaneously excited and their formation is accompanied by a narrow hysteresis. We show that bursting noises, stochastic resonance and deterministic chaos are determined by random transitions between these two dynamical states under slow and small changes of one of the model variables (z). We find that these effects take place even for such model parameters when hysteresis transforms into a step and they disappear when this step is smoothed out enough. We analyze some characteristics and conditions of formation of the deterministic chaos. We emphasize that such dynamical bistability and the effects related to it are universal phenomena and occur in a wide class of dynamical systems of different nature including brusselator.     

空间单位区域双二次有理贝赛尔曲面混沌特性研究

研究空间单位区域内两个二次曲面映射构成的函数的混沌特性, 发现了一种构造混沌的方法. 当一个曲面是单位区域内标准曲面, 另一个曲面随机生成时, 此函数是混沌的概率可以大于十分之一, 说明在满足一定条件时, 混沌是极其普遍的. 通过计算Lyapunov指数以及绘制分岔图等对该类函数的混沌特性进行分析, 根据参数变化的分岔图以及混沌曲面控制点的区域分布特性等寻找混沌映射函数, 得到了大量的二维混沌吸引子图形, 并对其中三个进行了详细研究. 另外, 把灰度图像作为离散二维函数, 首次研究了图像作为迭代表达式时表现出的一些混沌特性. 研究发现, 相同的或者相近的图像易于收敛到周期点上, 这个结果可以用于图像识别等研究领域. 关键词： 混沌 迭代 图像     

基于Laguerre多项式逼近法的随机双势阱Duffing系统的分岔和混沌研究

应用Laguerre正交多项式逼近法研究了含有随机参数的双势阱Duffing系统的分岔和混沌行为.系统参数为指数分布随机变量的非线性动力系统首先被转化为等价的确定性扩阶系统，然后通过数值方法求得其响应.数值模拟结果的比较表明，含有随机参数的双势阱Duffing系统保持着与确定性系统相类似的倍周期分岔和混沌行为，但是由于随机因素的影响，在局部小区域内随机参数系统的动力学行为会发生突变. 关键词： 双势阱Duffing系统 指数分布概率密度函数 Laguerre多项式逼近 随机分岔     

Bubbling transition to spatial mode excitation in an extended dynamical system

We investigated the transition to spatio-temporal chaos in spatially extended nonlinear dynamical systems possessing an invariant subspace with a low-dimensional attractor. When the latter is chaotic and the subspace is transversely stable we have a spatially homogeneous state only. The onset of spatio-temporal chaos, i.e. the excitation of spatially inhomogeneous modes, occur through the loss of transversal stability of some unstable periodic orbit embedded in the chaotic attractor lying in the invariant subspace. This is a bubbling transition, since there is a switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency. Hence the onset of spatio-temporal chaos depends critically both on the existence of a chaotic attractor in the invariant subspace and its being transversely stable or unstable.     

Noise can prevent onset of chaos in spatiotemporal population dynamics

Many theoretical approaches predict the dynamics of interacting populations to be chaotic but that has very rarely been observed in ecological data. It has therefore risen a question about factors that can prevent the onset of chaos by, for instance, making the population fluctuations synchronized over the whole habitat. One such factor is stochasticity. The so-called Moran effect predicts that a spatially correlated noise can synchronize the local population dynamics in a spatially discrete system, thus preventing the onset of spatiotemporal chaos. On the whole, however, the issue of noise has remained controversial and insufficiently understood. In particular, a well-built nonspatial theory infers that noise enhances chaos by making the system more sensitive to the initial conditions. In this paper, we address the problem of the interplay between deterministic dynamics and noise by considering a spatially explicit predator-prey system where some parameters are affected by noise. Our findings are rather counter-intuitive. We show that a small noise (i.e. preserving the deterministic skeleton) can indeed synchronize the population oscillations throughout space and hence keep the dynamics regular, but the dependence of the chaos prevention probability on the noise intensity is of resonance type. Once chaos has developed, it appears to be stable with respect to a small noise but it can be suppressed by a large noise. Finally, we show that our results are in a good qualitative agreement with some available field data.     

Chaos Generation in Opto-Mechanical Coupling System Assisted by Two-Level Atoms

Chaos is important in nonlinear science and promotes the development of fundamental studies, such as neural networks, extreme event statistics, and electron transport. In this paper, a theoretical scheme for generating dynamical chaos in a Fabry–Perot cavity with two-level atoms is investigated. Through the injection of atoms, controllable chaos of the cavity and mechanical oscillator is achieved simultaneously by the external laser field. The trajectory and the maximal Lyapunov exponent that characterize the properties of the chaos could be optimized by adjusting the coupling constant, driving field, injection of atoms, the frequency of the cavity, and the frequency of the mechanical oscillator. This study provides a new idea to manipulate the cavity and mechanical chaos assisted by two level atoms. It is hoped that the results presented can benefit controllable chaos generation and its applications.     

Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.     

用同步复极化终止心脏中的螺旋波和时空混沌

为了模拟电击除颤导致动作电位持续时间缩短, 在Luo-Rudy相I心脏模型中引入了同步复极化. 研究了同步复极化对螺旋波和时空混沌动力学的影响. 数值结果表明: 在控制周期比较小的情况下, 同步复极化可以有效消除螺旋波和时空混沌, 在有一些控制参数下, 同步复极化只能消除螺旋波, 或者只能消除时空混沌. 当螺旋波不被控制时, 观察到螺旋波转变为长周期和长波长的螺旋波或破碎成时空混沌的现象. 并对控制机制进行了分析. 关键词： 螺旋波 时空混沌 同步复极化 控制     

Experimental observation of wave chaos in a conventional optical resonator

We report on an experimental observation of optical wave chaos in a resonator consisting of three standard, high-reflectivity mirrors. The nonseparability of the wave equation necessary for chaos is introduced by violating the paraxial approximation. Until recently progress in optical wave chaos was hampered by the inherent difficulty in realizing suitable microscopic systems; now this novel, macroscopic approach offers complete and easy control and allows unprecedented study of optical wave chaos.     

FROM THREE-TORUS TO FRACTAL AND THEN TO CHAOS

We use numerical methods to study the model x(n+l)=λx(n) (x(n)-1)+ef(θ(n),φ(n)),θ(n+1)=θ(n)+A, φ(n+1)=φ(n)+B. As e is small, we get doubled three-tori. Increasing e the tori become fractal three-tori, the dimension is not integer, while the trajectory does not diverge exponentially. Finally it changes into chaos. The critical parameter values.for chaos are approximately calculated. So one of the roads from three-tori to chaos is three-tqri→fractal three-tori (not chaos)→chaos.     

Quantum Lyapunov Exponents

We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (quantum flow) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.