二维logistic映射的动力学行为和奇怪吸引子的分形特征

研究了二维logistic映射的动力学行为和奇怪吸引子的分形特征.利用分岔图、相图和Lyapunov指数谱分析系统的分岔过程,研究系统通向混沌的道路并确定系统处于混沌运动的参数区间;采用G-P算法计算奇怪吸引子的关联维数和Kolmogorov熵,对奇怪吸引子的分形特征定量刻画;采用逃逸时间算法构造奇怪吸引子的彩色广义M-J集,对奇怪吸引子的分形特征定性表征.结果表明,这些分析方法的配合使用可以更全面、形象地描述奇怪吸引子的分形特征.     

Effect of resonant-frequency mismatch on attractors

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.     

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.     

Determination of chaotic attractors in the rat brain

The existence of low-dimensional deterministic structures in experimental time series, derived from the occurrences of spikes in electrophysiological recordings from rat brains, has been revealed in 7 out of 27 samples. The correlation dimension of the chaotic attractors ranged between 0.14 and 3.3 embedded in a space of dimension 2–6. A test on surrogate data was also performed.     

Markov chains and discrete chaos

The connection between the statistics of a one-dimensional system exhibiting discrete chaos and its underlying deterministic law is formulated in terms of Markov chains. The correlation functions of piecewise linear Markovian maps are computed exactly using linear recursion formulae. An example of a non-Markovian map is given which can be approximated by Markovian maps.     

IFS吸引子的计算机模拟

阐述了迭代函数系(Iterated Function System,用IFS表示)理论及确定性算法。利用计算机对某一IFS的分形吸引子进行模拟,讨论了当参数变化时吸引子的变化规律;根据IFS的分维数定理,求出某些吸引子的分维数。     

Chaotic Phenomena in a Two-Photon Laser with Injected Signal

The instability and the chaotic phenomena in a two-photon laser with injected signal are discussed for the homogeneously broadened single mode ring cavity. The structure of the system's attractors is considered by using the Lyapunov exponents and the Lyapunov dimension. The strange attractors of chaos and superchaos are found. The strange attractor displaying superchaos is not observed in one-photon laser with injected signal.     

Nonlinear response and chaos in semiconductors induced by impact ionization

A review is given of the nonlinear response and chaos induced by impact ionization of neutral shallow donors, observed in n-GaAs. Two kinds of the observation are described; (i) firing wave instability, and (ii) periodically driven current filament. For the firing wave instability, several important aspects are discussed including the selective excitation of the current filaments and the deterministic nature of the firing density wave. The nonlinear response of a periodically driven current filament has been investigated by applying a dc+ac bias of the form ofV _dc+V _ac sin(2f ₀ t), wheref ₀1 MHz. The carrier dynamics and the bifurcation routes to chaos are discussed in terms of the observed phase diagram and the bifurcation map. The deterministic nature of the strange attractors are described in detail in terms of the correlation dimension and the Kolmogorov entropy.     

横掠管束周期性充分发展对流换热的混沌分析

本文利用混沌理论分析了横掠管束周期性充分发展对流换热的非稳定性问题,即通过速度U的时间序列的重构相空间计算出关联维数D2,并通过时间序列分析了该非线性动力系统的功率谱特性。分析结果表明,本文所研究的横掠管束周期性充分发展对流换热系统在所给出的控制参数Re=937.7下出现的非稳定性问题属于混沌现象。系统的整体状态可用奇怪吸引子来描述,当延迟时间选择为5,该时间序列的重构相空间的嵌入维数增至5时,该吸引子的分维数趋于定值1.63。     

Controlling chaos and deterministic directed transport in inertia ratchets using backstepping control

We address the problem of controlling chaotic motion and deterministic directed transport in inertia ratchets. We employ a recursive backstepping nonlinear control technique to control intermittent chaos and then track a desired trajectory by means of the same technique. For the parameter regime where two non-identical attractors coexist in phase space, we propose a new backstepping control scheme that is capable of controlling the directed transport exhibited by these attractors. Numerical simulations show that the controllers are singularity free and the closed-loop systems are globally stable.     

Control of current reversal in single and multiparticle inertia ratchets

We have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of the amplitude of the external driving force. Two experimentally inspired methods are used. In the first method, the same initial condition is used for each new value of the amplitude. In the second method, the last position and velocity is used as the new initial condition when the amplitude is changed. The two methods are found to be complementary for control of current reversal, because the first one elucidates the existence of different attractors and gives information about their basins of attraction, while the second method, although history dependent, shows the locking process. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. An unlocking effect is produced when a chaos to order transition limits the control range.     

Maximally complex simple attractors

A relatively small number of mathematically simple maps and flows are routinely used as examples of low-dimensional chaos. These systems typically have a number of parameters that are chosen for historical or other reasons. This paper addresses the question of whether a different choice of these parameters can produce strange attractors that are significantly more chaotic (larger Lyapunov exponent) or more complex (higher dimension) than those typically used in such studies. It reports numerical results in which the parameters are adjusted to give either the largest Lyapunov exponent or the largest Kaplan-Yorke dimension. The characteristics of the resulting attractors are displayed and discussed.     

On the evidence of deterministic chaos in ECG: Surrogate and predictability analysis

The question whether the human cardiac system is chaotic or not has been an open one. Recent results in chaos theory have shown that the usual methods, such as saturation of correlation dimension D(2) or the existence of positive Lyapunov exponent, alone do not provide sufficient evidence to confirm the presence of deterministic chaos in an experimental system. The results of surrogate data analysis together with the short-term prediction analysis can be used to check whether a given time series is consistent with the hypothesis of deterministic chaos. In this work nonlinear dynamical tools such as surrogate data analysis, short-term prediction, saturation of D(2) and positive Lyapunov exponent have been applied to measured ECG data for several normal and pathological cases. The pathology presently studied are PVC (Premature Ventricular Contraction), VTA (Ventricular Tachy Arrhythmia), AV (Atrio-Ventricular) block and VF (Ventricular Fibrillation). While these results do not prove that ECG time series is definitely chaotic, they are found to be consistent with the hypothesis of chaotic dynamics. (c) 1998 American Institute of Physics.     

旋转对称的广义Lorenz奇怪吸引子

阐述了计算微分方程组最大Lyapunov指数的技术,介绍了由一维可观察量计算系统关联维数的方法.利用Lyapunov指数作判据,通过坐标变换,构造了具有旋转对称性的广义Lorenz奇怪吸引子,分析了奇怪吸引子的运动特征并计算了奇怪吸引子的关联维数.     

Period-Doubling Cascades and Strange Attractors in Extended Duffing-Van der Pol Oscillator

The dynamical behavior of the extended Duffing-Van der Pol oscillator is investigated numerically in detail. With the aid of some numerical simulation tools such as bifurcation diagrams and Poinearé maps, the different routes to chaos and various shapes of strange attractors are observed. To characterize chaotic behavior of this oscillator system, the spectrum of Lyapunov exponent and Lyapunov dimension are also employed.     

Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors

The Kuramoto-Sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films. They are characterized by the coexistence of coherent spatial structures with temporal chaos. We investigate some global dynamical properties, including nonlinear stability. We demonstrate their low modal behavior, in terms of determining modes; and that the fractal dimension of all attractors is bounded by a universal constant times $\text{≈L}{}^{\mathrm{<mtext>13</mtext><mtext>8</mtext>}}$, where ≈L is a dimensionless pattern cell size (in the one-dimensional case). Such equations are indeed a paradigm of low-dimensional behavior for infinite-dimensional systems.     

Control of dynamical systems behavior by parametric perturbations: An analytic approach

The problem of parametric suppression of deterministic chaos is considered. It is proved that certain parametric perturbations of a one-dimensional map with chaotic dynamics can lead to a transition of that map into a regime of regular behavior.     

Low dimensional chaos in the AT and GC skew profiles of DNA sequences

This paper investigates the existence of low-dimensional deterministic chaos in the AT and GC skew profiles of DNA sequences. It has taken DNA sequences from eight organisms as samples. The skew profiles are analysed using continuous wavelet transform and then nonlinear time series methods. The invariant measures of correlation dimension and the largest Lyapunov exponent are calculated. It is demonstrated that the AT and GC skew profiles of these DNA sequences all exhibit low dimensional chaotic behaviour. It suggests that chaotic properties may be ubiquitous in the DNA sequences of all organisms.     

Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes

This paper is a review of the present status of studies relating to occurrence of deterministic chaos and its characterization in one-dimensional maps. As our primary aim is to introduce the nonspecialists into this fascinating world of chaos we start from very elementary concepts and give sufficient arguments for clarity of ideas. The two main scenarios during onset of chaos viz. the period doubling and intermittency are dealt with in detail. Although the logistic map is often discussed by way of illustration, a few more interesting maps are mentioned towards the end.     

Correlation Dimension

In many situations, both deterministic and probabilistic, one can develop further the study of the multifractal structure of a dynamical system, particularly when there exist strange attractors. Multifractal refers to a notion of size emphasizing the variations of the weigth of the measure. In such schemes, one has to compute a free energy function associated to some sequence of partitions. We relate the free energy function, associated to a sequence of uniform partitions of exponentially decreasing diameters, and the correlation dimension which refers to a quantity that is the most accessible in numerical computations. Finally we discuss of two assumptions for the existence of free energy functions.