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.     

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.     

On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

For Hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t, and all numbers of particles N. This proof was generalized by Wojtkowski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.     

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.     

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.     

The Lyapunov spectrum of a continuous product of random matrices

We present a functional integration method for the averaging of continuous productsP _t ofN×N random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum ofP _t . This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white-noise force field.     

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.      

Finite-Time Lyapunov Exponents for Products of Random Transformations

I show how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decomposition into angular, diagonal, and shear degrees of freedom. In the case of a Gaussian process a set of variables, adapted to the Iwasawa decomposition and still having a Gaussian distribution, is introduced and used to compute the statistics of the finite-time Lyapunov spectrum of the process. The variables also allow to show the exponential freezing of the shear degrees of freedom, which contain information about the Lyapunov eigenvectors.     

When Do Tracer Particles Dominate the Lyapunov Spectrum?

Dynamical instability is studied in a deterministic dynamical system of Hamiltonian type composed of a tracer particle in a fluid of many particles. The tracer and fluid particles are hard balls (disks, in two dimensions, or spheres, in three dimensions) undergoing elastic collisions. The dynamical instability is characterized by the spectrum of Lyapunov exponents. The tracer particle is shown to dominate the Lyapunov spectrum in the neighborhoods of two limiting cases: the Lorentz-gas limit in which the tracer particle is much lighter than the fluid particles and the Rayleigh-flight limit in which the fluid particles have a vanishing radius and form an ideal gas. In both limits, a gap appears in the Lyapunov spectrum between the few largest Lyapunov exponents associated with the tracer and the rest of the Lyapunov spectrum.     

Multidimensional mappings in antiferromagnetic models and Lyapunov exponents with the magnetization plateau

We consider the Heisenberg model with two- and three-spin exchange interactions on a recursive ladder in a strong magnetic field. Recurrent relations for branches of the partition function of the Ising model with two- and three-spin exchange interactions are deduced. As a recursive lattice the zigzag ladder is chosen. In the antiferromagnetic case magnetization plateau are observed at low temperatures. Lyapunov exponents for the three-dimensional mapping at low temperatures are calculated. It is shown that for some values of two- and three-spin exchange parameters in the antiferromagnetic case the maximum of the Lyapunov exponent approaches zero.     

Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.     

Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

We study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices without any particular structure the generalized Lyapunov exponents become equal in this limit and the value of one of the generalized Lyapunov exponents is obtained by simple arguments. On the contrary, for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized Lyapunov exponents remains different forD , indicating that high dimensionality cannot always destroy intermittency.     

Non-Gaussian Fluctuations of Local Lyapunov Exponents at Intermittency

In intermittent dynamical systems, the distributions of local Lyapunov exponents are markedly non-Gaussian and tend to be asymmetric and fat-tailed. A comparative analysis of the different time-scales in intermittency provides a heuristic explanation for the origin of the exponential tails, for which we also obtain an analytic expression deriving from a more quantitative theory. Application is made to several examples of discrete dynamical systems displaying intermittent dynamics.     

Control of chaos in permanent magnet synchronous motor by using optimal Lyapunov exponents placement

Permanent Magnet Synchronous Motor (PMSM) experiences chaotic behavior for a certain range of its parameters. In this case, since the performance of the PMSM degrades, the chaos should be eliminated. In this Letter, the control of the undesirable chaos in PMSM using Lyapunov exponents (LEs) placement is proposed that is also improved by choosing optimal locations of the LEs in the sense of predefined cost function. Moreover, in order to provide the physical realization of the method, nonlinear parameter estimator for the system is suggested. Finally, to show the effectiveness of the proposed methodology, the simulation results for applying this control strategy are provided.     

Weak disorder expansion of Lyapunov exponents of products of random matrices: A degenerate theory

The weak disorder expansion of Lyapunov exponents of products of random matrices is derived by a new method. Our treatment can be easily generalized to the problem when in the limit of zero randomness two eigenvalues of the matrices are equal. For real degenerate matrices, the formula for the leading term of the Lyapunov exponent is derived. It has the form of a continuous fraction, which converges quickly to the exact value.     

Rigorous bounds of the Lyapunov exponents of the one-dimensional random Ising model

We find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations.     

Partial Hyperbolicity, Lyapunov Exponents and Stable Ergodicity

We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.     

The Lyapunov spectrum as the Newton method

For a class of dynamical systems, the cookie-cutter maps, we prove that the Lyapunov spectrum coincides with the map given by the Newton–Raphson method applied to the derivative of the pressure function.     

Lyapunov exponents of stochastic dynamical systems

It is shown that stochastic equations can have stable solutions. In particular, there exists stochastic dynamics for which the motion is both ergodic and stable, so that all trajectories merge with time. We discuss this in the context of Monte Carlo-type dynamics, and study the convergence of nearby trajectories as the number of degrees of freedom goes to infinity and as a critical point is approached. A connection with critical slowdown is suggested.     

精确配置离散动力系统的所有Lyapunov指数

针对给定的确定性离散时间动力学系统,提出一种可以精确配置受控系统所有Lyapunov指数的混沌控制与反控制新方法.本文不限制给定系统的具体形式,仅利用取模运算和预设的一组Lyapunov指数,设计出一种新的状态反馈控制器.在控制器的作用下,受控系统的Lyapunov指数与预设的Lyapunov指数一致.控制器的形式简单,可以用于解决混沌控制和混沌反控制两种问题.给出了数学证明和两个实例,仿真结果显示了算法的有效性.