有限温度下Na_4 团簇赝转动的局域Lyapunov指数研究

:在紧束缚分子动力学模型基础上,对具有D2h对称性的Na4团簇在不同温度T=76K、112K、212K和224K下的热力学行为进行了研究。根据局域Lyapunov指数分布的变化,对由温度T=76K、112K向温度T=212K、224K的变化中会出现的赝转动所表现的混沌现象进行了分析     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Correlation regimes in systems of soft spherical particles studied by their Lyapunov exponents

We performed molecular dynamics simulations of soft spherical particles over wide ranges of densities and temperatures corresponding to fluids, glasses and crystalline solids, and calculated the full Lyapunov spectra of these systems. For either phase corresponding exponents essentially scale with the square-root of the temperature in accordance with kinetic theory. The density dependence is more pronounced and less systematic. The shape of the spectrum of a glass is different from that of a crystalline solid at the same density and temperature and resembles the spectrum of the initial dense liquid-like phase. For dilute gases the sum of the positive exponents approaches zero and is increasingly dominated by the largest exponent. Although systematic changes of the Lyapunov spectra were observed, it seems that the spectral shape does not uniquely determine the phase of the system.     

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.     

Symbolic dynamics and topological entropy at the onset of pruning

The topological entropy and pruning rules are investigated for two-dimensional smooth maps at the onset of pruning. Typically the difference of the parameter-dependent topological entropy from its maximum value increases with a power law. Superimposed on this decrease, there are periodic or quasiperiodic oscillations on a logarithmic scale. Both, the scaling exponent and the periodicity are determined by the Lyapunov exponents of the first pruned orbit and the minimal number of letters in the alphabet of the symbolic dynamics. If, at the onset of pruning, the averaged Lyapunov exponent is sufficiently large and the first pruned orbit is homoclinic, the entropy function of area-preserving maps exhibits a series of plateaux. On the plateaux, the symbolic dynamics can be described by finitely many finite forbidden words. There is a series of plateaux which, in different systems, can be described by the same type of forbidden words.     

Sensitivity to initial conditions and lyapunov exponent of a quasiperiodic system

The dynamics of a quasiperiodic map is analyzed both in the presence and in the absence of weak noise. It is shown that, in the presence of weak noise, a strange chaotic attractor with a negative Lyapunov exponent and sensitive dependence of trajectories on the initial conditions can exist in the system. This means that the types of motion of a fluctuating system cannot be classified only by the sign of the leading Lyapunov exponent.     

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.     

A finite-time exponent for random Ehrenfest gas

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit.     

切换系统Lyapunov指数的算法及应用

Lyapunov指数是判定系统非线性行为的重要工具,然而目前的大多算法并不适用于切换系统.在传统Jacobi法的基础上,提出了一种新算法,可以直接计算得到n维切换系统的n个Lyapunov指数.首先,根据切换面处相邻轨线的动态变化规律,从相空间几何推导出切换面处轨线变化的Jacobi矩阵;然后,对该矩阵进行QR分解,从而利用R的对角线元素实现Lyapunov指数的切换补偿;最后,将新算法应用到平面双螺旋混沌系统、Glass网络和航天器供电系统三个实例中,并将计算结果与Poincaré映射方法的计算结果进行比较,对新算法的有效性进行验证.     

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.     

Correlation effects of two interacting particles in a circular billiard

The maximal Lyapunov exponent is determined numerically for two classical unequal-mass interacting particles inside a circular billiard and subjected to a static magnetic field. A Yukawa potential is used for the interaction between the particles. Transitions from short to long interaction ranges and from equal to infinite mass ratio between particles are discussed. Correlations effects between particles strongly determine the dynamics inside the billiard. A qualitative change in the Lyapunov exponent dependence on the interaction range between particles is observed by the transition from weak to strong couplings. Poincaré surfaces of section are also used to describe the dynamics in the limit of infinite mass ratio.     

Largest Lyapunov exponent in molecular systems. II: Quaternion coordinates and application to methane clusters

We present a numerical procedure for extracting Lyapunov characteristic exponents from classical molecular-dynamics simulations of molecular systems. The theoretical frame chosen to describe the orientational degrees of freedom is the quaternions scheme. We apply the method to small methane clusters. Two different model potentials are used to investigate the role of internal molecular motion on the nonlinear dynamics, and several parameters are calculated to study the thermodynamics and chaotic dynamics of these clusters. Evidence is found for a solidlike to plasticlike phase transition occurring with the release of the orientational degrees of freedom, at low temperatures below the melting point. The largest Lyapunov exponent increases significantly during this transition, but it exhibits no particular variation during melting.     

Long chaotic transients in complex networks

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulse-coupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and vary by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transients exhibit a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically.     

激发核系统中最大Lyapunov指数、密度涨落以及多重碎裂之间的关系

基于量子分子动力学模型,系统地研究了从48Ca到298114一系列核素在不同温度下的最大Lyapunov指数、密度涨落以及体系多重碎裂之间的关系.发现最大Lyapunov指数随温度变化有一峰值出现(该峰值所对应的温度为"临界温度"),在该临界温度时体系的密度涨落达到最大,碎块的质量分布能够给出较好的PowerLaw指数.通过对最大Lyapunov指数与密度涨落随时间变化行为的研究,发现密度涨落的时间尺度要大于混沌的时间尺度,意味着混沌的概念可以用来研究体系的多重碎裂过程.最后还给出了有限体系相变的临界温度随体系大小变化的规律. Within a quantum molecular dynamics model we calculate the largest Lyapunov exponent (LLE), the density fluctuation, and the mass distribution of fragments for a series of nuclear systems at different initial temperatures. It is found that the LLE peaks at the temperature ("critical temperature") where the density fluctuation reaches a maximal value and the mass distribution fragments is fitted best by the Fisher s power law from which the critical exponents for mass and charge distribution are obtain...     

Measuring topological chaos

The orbits of fluid particles in two dimensions effectively act as topological obstacles to material lines. A spacetime plot of the orbits of such particles can be regarded as a braid whose properties reflect the underlying dynamics. For a chaotic flow, the braid generated by the motion of three or more fluid particles is computed. A "braiding exponent" is then defined to characterize the complexity of the braid. This exponent is proportional to the usual Lyapunov exponent of the flow, associated with separation of nearby trajectories. Measuring chaos in this manner has several advantages, especially from the experimental viewpoint, since neither nearby trajectories nor derivatives of the velocity field are needed.     

Lyapunov generation of entanglement and the correspondence principle

We show how a classically vanishing interaction generates entanglement between two initially nonentangled particles, without affecting their classical dynamics. For chaotic dynamics, the rate of entanglement is shown to saturate at the Lyapunov exponent of the classical dynamics as the interaction strength increases. In the saturation regime, the one-particle Wigner function follows classical dynamics better and better as one goes deeper and deeper in the semiclassical limit. This demonstrates that quantum-classical correspondence at the microscopic level does not require coupling to a large number of external degrees of freedom.     

Time Series Prediction Based on Chaotic Attractor

A new prediction technique is proposed for chaotic time series. The usefulness of the technique is that it can kick off some false neighbor points which are not suitable for the local estimation of the dynamics systems. A time-delayed embedding is used to reconstruct the underlying attractor, and the prediction model is based on the time evolution of the topological neighboring in the phase space. We use a feedforward neural network to approximate the local dominant Lyapunov exponent, and choose the spatial neighbors by the Lyapunov exponent. The model is tested for the Mackey-Glass equation and the convection amplitude of lorenz systems. The results indicate that this prediction technique can improve the prediction of chaotic time series.     

原子核液气相变的动力学研究

应用量子分子动力学模型,从原子核发生碎化时的碎块质量分布、密度涨落以及混沌动力学中描述混沌程度的最大Lyapunov指数等3个方面,对原子核的液气相变及其在临界点附近的行为进行较为全面的探讨.通过研究,发现在临界温度附近原子核发生最大的液气共存、密度涨落达到极大以及产生最大混沌构型.     

Time Series Prediction Based on Chaotic Attractor

A new prediction technique is proposed for chaotic time series. The usefulness of the technique is thatit can kick off some false neighbor points which are not suitable for the local estimation of the dynamics systems. Atime-delayed embedding is used to reconstruct the underlying attractor, and the prediction model is based on the timeevolution of the topological neighboring in the phase space. We use a feedforward neural network to approximate thelocal dominant Lyapunov exponent, and choose the spatial neighbors by the Lyapunov exponent. The model is testedfor the Mackey-Glass equation and the convection amplitude of lorenz systems. The results indicate that this predictiontechnique can improve the prediction of chaotic time series.     

Synchronization of self-sustained oscillators by common white noise

We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators.