Lyapunov Modes in Hard-Disk Systems

We consider simulations of a two-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exponents. To this spectrum are associated eigen-directions, called Lyapunov modes. We carefully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics of the modes (where it exists) follow linear laws, whose coefficients only depend on the density of the gas, but not on aspect ratio and very little on the boundary conditions.     

Hydrodynamic Lyapunov Modes in Translation-Invariant Systems

We study the implications of translation invariance on the tangent dynamics of extended dynamical systems, within a random matrix approximation. In a model system, we show the existence of hydrodynamic modes in the slowly growing part of the Lyapunov spectrum, which are analogous to the hydrodynamic modes discovered numerically by Dellago, Posch, and Hoover. The hydrodynamic Lyapunov vectors lose the typical random structure and exhibit instead the structure of weakly perturbed coherent long-wavelength waves. We show further that the amplitude of the perturbations vanishes in the thermodynamic limit, and that the associated Lyapunov exponents are universal.     

Lyapunov Modes of Two-Dimensional Many-Body Systems; Soft Disks, Hard Disks, and Rotors

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {}, its associated eigenvectors {}, and the time-averaged spectrum {}. Each local Lyapunov exponent describes the degree of instability associated with a well-defined direction—given by the associated unit vector —in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the vectors, all with relatively-small values of the time-averaged exponent , correspond to quite well-defined long-wavelength modes. We investigate soft particles from the same viewpoint here, and find no convincing evidence for corresponding modes. The situation is similar—no firm evidence for modes—in a simple two-dimensional lattice-rotor model. We believe that these differences are related to the form of the time-averaged Lyapunov spectrum near =0.     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.     

Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems

In the study of chaotic behaviour of systems of many hard spheres, Lyapunov exponents of small absolute values exhibit interesting characteristics leading to speculations about connections to non-equilibrium statistical mechanics. Analytical approaches to these exponents so far can be divided into two groups, macroscopically oriented approaches, using kinetic theory or hydrodynamics, and more microscopically oriented random-matrix approaches in quasi-one-dimensional systems. In this paper, I present an approach using random matrices and weak-disorder expansion in an arbitrary number of dimensions. Correlations between subsequent collisions of a particle are taken into account. It is shown that the results are identical to those of a previous approach based on an extended Enskog equation. I conclude that each approach has its merits, and provides different insights into the approximations made, which include the Stoßzahlansatz, the continuum limit, and the long wavelength approximation. The comparison also gives insight into possible connections between Lyapunov exponents and fluctuations.     

Lyapunov control of squeezed noise quantum trajectory

The quantum trajectory renders the optimal estimation of quantum state. It is a classical Itô stochastic differential equation. The Lyapunov global stabilization problem is solved for squeezed noise quantum trajectory. Lyapunov control stabilizes the quantum system toward one eigenstate. A two-level bistable quantum system is simulated as an example.     

Kirchhoff方程的相对常值特解及其Lyapunov稳定性

对于超细长弹性杆静力学的Kirchhoff方程，用动力学的概念和方法研究其常值特解 和稳定性问题.计算了Kirchhoff方程相对固定坐标系、截面主轴坐标系以及中心线Frenet 坐标系的常值特解，进行了Kirchhoff动力学比拟，用一次近似理论分别讨论了它们的Lyapu nov稳定性，导出了若干稳定性判据，并在参数平面上绘出了稳定域. 关键词： 超细长弹性杆 Kirchhoff方程 常值特解 Lyapunov稳定性     

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.     

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...     

Positive Lyapunov exponents in the Kramers oscillator

The maximum Lyapunov exponent is computed numerically for the double-well oscillator in a heat bath. Positive exponents are found in a wide range of friction coefficients in the low-damping regime.     

Bistability in Coupled Oscillators Exhibiting Synchronized Dynamics

We report some new results associated with the synchronization behavior of two coupled double-well Duffing oscillators （DDOs）. Some sufficient algebraic criteria for global chaos synchronization of the drive and response DDOs via linear state error feedback control are obtained by means of Lyapunov stability theory. The synchronization is achieved through a bistable state in which a periodic attractor co-exists with a chaotic attractor. Using the linear perturbation analysis, the prevalence of attractors in parameter space and the associated bifurcations are examined. Subcritical and supercritical Hopf bifurcations and abundance of Arnold tongues -- a signature of mode locking phenomenon are found.     

Positivity of Lyapunov Exponents for a Continuous Matrix-Valued Anderson Model

We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all energies in (2, +∞) except those in a discrete set, which leads to absence of absolutely continuous spectrum in (2, +∞). This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.      

Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

S-盒的Lyapunov指数研究

在密码算法的设计中,S-盒有着信息混淆的重要功能.传统的S-盒的密码学指标一般包括线性偏差、差分特征、代数免疫度、不动点个数、雪崩效应等.2006年,Kocarev给出了有限集合上的离散混沌理论.本文借鉴该理论,在汉明距离的基础上给出了S-盒的Lyapunov指数的定义,利用该定义计算了几个密码算法中的S-盒的Lyapunov指数值,并进行了比较.证明了在欧氏距离上定义的Lvapunov指数最大的映射,按本文提出的S-盒的Lyapunov指数的定义其Lyapunov指数为0;讨论了S-盒的Lyapunov指数与S-盒的雪崩效应之间的关系,该关系实际上是混沌理论中的蝴蝶效应与密码学中的雪崩效应之间的关系.本文提出的S-盒的Lyapunov指数的定义可视为对传统的S-盒的密码学指标的补充.     

Nonlinear finite-time Lyapunov exponent and predictability

In this Letter, we introduce a definition of the nonlinear finite-time Lyapunov exponent (FTLE), which is a nonlinear generalization to the existing local or finite-time Lyapunov exponents. With the nonlinear FTLE and its derivatives, the limit of dynamic predictability in large classes of chaotic systems can be efficiently and quantitatively determined.     

The Lyapunov Spectrum Is Not Always Concave

We characterize one-dimensional compact repellers having non-concave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra. The first author was partially supported by Proyecto Fondecyt 11070050. Both authors were partially supported by Research Network on Low Dimensional Systems, PBCT/CONICYT, Chile.     

A differential algorithm for the Lyapunov spectrum

We present a new algorithm for computing the Lyapunov exponents spectrum based on a matrix differential equation. The approach belongs to the so-called continuous type, where the rate of expansion of perturbations is obtained for all times, and the exponents are reached as the limit at infinity. It does not involve exponentially divergent quantities so there is no need of rescaling or realigning of the solution. We show the algorithm’s advantages and drawbacks using mainly the example of a particle moving between two contracting walls.     

Upper Quantum Lyapunov Exponent and Anosov Relations for Quantum Systems Driven by a Classical Flow

We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties.     

Symmetry of Lyapunov spectrum

The symmetry of the spectrum of Lyapunov exponents provides a useful quantitative connection between properties of dynamical systems consisting ofN interacting particles coupled to a thermostat, and nonequilibrium statistical mechanics. We obtain here sufficient conditions for this symmetry and analyze the structure of 1/N corrections ignored in previous studies. The relation of the Lyapunov spectrum symmetry with some other symmetries of dynamical systems is discussed.