Decoherence and the Loschmidt echo

Decoherence causes entropy increase that can be quantified using, e.g., the purity sigma=Trrho(2). When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo M(t). It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of sigma(t) and the average Mmacr;(t). In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of sigma and Mmacr; has a regime where it is dominated by the Lyapunov exponents.     

Correlation decay in quantum chaotic billiards with bulk or surface disorder

We study the decay properties of correlation functions in quantum billiards with surface or bulk disorder. The quantum system is modeled by means of a tight-binding Hamiltonian with diagonal disorder, solved on LxL clusters of the square lattice. The correlation function is calculated by launching the system at t=0 into a wave function of the regular (clean) system and following its time evolution. The results show that the correlation function decays exponentially with a characteristic correlation time (inverse of the Lyapunov exponent lambda). For small enough disorder the Lyapunov exponent is approximately given by the imaginary part of the self-energy induced by disorder. On the other hand, if the scaling of the Lyapunov exponent with L is investigated by keeping constant l/L, where l is the mean free path, the results show that lambda is proportional to 1/L.     

Faster than Lyapunov decays of the classical Loschmidt echo

We show that in the classical interaction picture the echo dynamics, namely, the composition of perturbed forward and unperturbed backward Hamiltonian evolution, can be treated as a time-dependent Hamiltonian system. For strongly chaotic (Anosov) systems we derive a cascade of exponential decays for the classical Loschmidt echo, starting with the leading Lyapunov exponent, followed by a sum of the two largest exponents, etc. In the loxodromic case a decay starts with the rate given as twice the largest Lyapunov exponent. For a class of perturbations of symplectic maps the echo dynamics exhibits a drift resulting in a superexponential decay of the Loschmidt echo.     

Chaos in low-dimensional hamiltonian maps

Symplectic maps with more than two degrees of freedom constructed by coupling N area-preserving Chiricov-Taylor standard maps are investigated by numerical methods. We find the asymptotic (for N→∞) distribution of the N positive Lyapunov exponents which is attained already for surprisingly small N. To test the errors in calculating Lyapunov exponents from finite parts of trajectories we calculate the fluctuations of the effective Lyapunov exponents as a function of the number of iterations and find a nontrivial decay on time scales decreasing with increasing degree of freedom. These fluctuations are due to clinging of trajectories to regular orbits.     

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.     

When can one observe good hydrodynamic Lyapunov modes?

Inspired by recent results on differences in fluctuations of finite-time Lyapunov exponents between hard-core and soft-potential systems we surmise that partial domination of the Oseledec splitting (DOS) with respect to subspaces associated with near-zero Lyapunov exponents is essential for observing good hydrodynamic Lyapunov modes (HLMs). Numerical results for coupled map lattices are presented to show the importance of DOS for observing good HLMs. This is achieved by relating splitting parameters to the maximum value of the Lyapunov mode structure factor.     

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.     

Arnold's method for asymptotic stability of steady inviscid incompressible flow through a fixed domain with permeable boundary

The flow of an ideal fluid in a domain with a permeable boundary may be asymptotically stable. Here the permeability means that the fluid can flow into and out of the domain through some parts of the boundary. This permeability is a principal reason for the asymptotic stability. Indeed, the well-known conservation laws make the asymptotic stability of an inviscid flow impossible, if the usual no flux condition on a rigid wall (or on a free boundary) is employed. We study the stability problem using the direct Lyapunov method in the Arnold's form. We prove the linear and nonlinear Lyapunov stability of a two-dimensional flow through a domain with a permeable boundary under Arnold's conditions. Under certain additional conditions, we amplify the linear result and prove the exponential decay of small disturbances. Here we employ the plan of the proof of the Barbashin-Krasovskiy theorem, established originally only for systems with a finite number of degrees of freedom. (c) 2002 American Institute of Physics.     

Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals

In this paper, the localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. Numerical results of the localization lengths of SH-wave are presented and discussed in ordered and disordered piezoelectric phononic crystals to identify the different effect degrees for the decay of electrical potential in the polymers and the randomness on the localization level. For the disordered case, disorder in the thickness of the polymers and disorder in the elastic constant of the piezoelectric ceramics are all considered. The results show that some parameters such as the incident angle of elastic wave, the randomness degree and the piezoelectricity of piezoelectric ceramics and so on have pronounced effects on the frequency-dependent localization length.     

Stretching in a model of a turbulent flow

Using a multi-scaled, chaotic flow known as the KS model of turbulence [J.C.H. Fung, J.C.R. Hunt, A. Malik, R.J. Perkins, Kinematic simulation of homogeneous turbulence by unsteady random fourier modes, J. Fluid Mech. 236 (1992) 281-318], we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.     

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.     

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

To identify and to explain coupling-induced phase transitions in coupled map lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number $N$ of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for $N=3$ and numerically for $N\ge 3$ , that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.     

Chaotic itinerancy generated by coupling of Milnor attractors

We report the existence of chaotic itinerancy in a coupled Milnor attractor system. The attractor ruins consist of tori or local chaos generated from the original Milnor attractors. The chaotic behavior exhibited by a single orbit can be considered a "nonstationary" state, due to the extremely slow convergence of the Lyapunov exponents, but the behavior averaged over randomly chosen initial conditions is consistent with the limit theorem. We present as a possibly new indication of chaotic itinerancy the presence of slow decay of large fluctuations of the largest Lyapunov exponent.     

Spectral properties of chimera states

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.     

Time-oscillating Lyapunov modes and the momentum autocorrelation function

The Lyapunov vectors corresponding to the steps of Lyapunov spectra for many-particle systems show time-oscillating behavior in two types of Lyapunov modes, one associated with time-translational invariance and the other with spatial translational invariance. Our result is that, for each coordinate direction, the longest period of the Lyapunov modes is twice as long as the period of the momentum autocorrelation function. A simple explanation for this relation is proposed and we argue that this result is generally true for many-particle systems. This gives the first quantitative connection between the Lyapunov modes and an experimentally accessible quantity.     

Global stability analysis of fluid flows using sum-of-squares

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems.We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.     

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.     

Decay of the Loschmidt echo for quantum states with sub-planck-scale structures

Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than variant Planck's over 2pi [W. H. Zurek, Nature (London) 412, 712 (2001)]]. We investigate the influence of those sub-Planck-scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wave packet, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interference.     

Geodesic Stability for Kehagias-Sfetsos Black Hole in Hořava-Lifshitz Gravity via Lyapunov Exponents

By computing the Lyapunov exponent, which is the inverse of the instability time scale associated with this geodesic motion we show that for a general Kehagias-Sfetsos (KS) solution, there is two region of space which in both of them the equatorial timelike geodesics are stable via Lyapunov measure of stability.     

Orbit Tracking of Quantum State for Non-Markovian Quantum System Based on Model Reference Adaptive Control

Utilizing model reference adaptive control theory and Lyapunov stability theorem, we derive the adaptive law for the model reference adaptive system. Then we design the Lyapunov control law by double control functions and investigate the orbit tracking of quantum state for non-Markovian quantum system with phase relaxation and energy dissipative relaxation. The influence of Ohmic reservoir with Lorentz-Drude regularization is numerically studied for a two-level system. The simulations show that the controlled quantum system will track the target orbit with a small oscillation due to the non-Markovian environmental memory effect, which indicates the orbit tracking of non-Markovian quantum system is incomplete.