Renormalization group analysis of some dynamical systems with noise

We formulate a renormalization group analysis for the study of the accumulation of period doubling in the presence of noise. The main tool is a renormalization of the time evolution of the noise. The critical indices depend on the nature of the noise, but are given by thermodynamic quantities describing the large deviations of the Lyapunov exponent of the linearized random renormalization.     

Master equations with memory for systems driven by classical noise

A non-Markovian master equation is obtained for a two level atom driven by a phase noisy laser. The derivation is based on obtaining an equation for the density operator of the system averaged over the previous histories of the external noise. Averaging over the current value of the noise variable by means of the Zwanzig-Nakajima projection operator technique leads to a master equation with memory and a local-in-time master equation. The solutions to the resultant non-Markovian master equation, the structural properties of the equation, and the amenability of the equation to unravelling by the quantum trajectory method are all investigated.     

On a simple derivation of master equations for diffusion processes driven by white noise and dichotomic Markov noise

A very simple way is presented of deriving the partial differential equations (the master equations) satisfied by the probability density for certain kinds of diffusion processes in one dimension, in which the driving term is a Gaussian white noise, or a dichotomic noise, or a combination of the two. The method involves the use of certain ‘formulas of differentiation’ to derive the equations obeyed by the characteristic functions of the processes concerned, and thence the corresponding master equations. The examples presented cover a substantial number of diffusion processes that occur in physical modelling, including some master equations derived recently in the literature for generalizations of persistent diffusion.     

Almost-sure asymptotic stability of a general four-dimensional system driven by real noise

In the first part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the rotation numbers for a general four-dimensional dynamical system driven by a small-intensity real noise process. Stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. This work is an extension of the work of Sri Namachchivaya and Van Roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered.     

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.     

准单色噪声驱动的耗散动力系统的信息熵演化

准单色噪声是一类所谓真正有色的噪声.本文对准单色噪声驱动的耗散动力系统的信息熵演化进行研究,文中首先运用线性变换的方法给出了所研究系统的Fokker-Planck方程,然后结合Shannon信息熵定义推导了该系统随时间演化信息熵的精确表达式, 最后分析了系统耗散参数和准单色噪声对系统信息熵的显著影响.     

准单色噪声驱动的耗散动力系统的信息熵演化

准单色噪声是一类所谓真正有色的噪声.本文对准单色噪声驱动的耗散动力系统的信息熵演化进行研究,文中首先运用线性变换的方法给出了所研究系统的Fokker-Planck方程,然后结合Shannon信息熵定义推导了该系统随时间演化信息熵的精确表达式,最后分析了系统耗散参数和准单色噪声对系统信息熵的显著影响.     

On certain time characteristics of dynamical systems driven by noise

The moments of the transition time of non-linear dynamical systems driven by noise are introduced. It is demonstrated that the well-known recurrence formula for the moments of the first passage time (FTP) of the absorbing boundary by a Brownian particle can also be used to obtain the exact values of the moments of the transition time in non-linear dynamical systems with noise, described by a symmetric potential profile.     

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.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise

The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied.The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.     

Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on $L^2(\mathbb R)\otimes \mathbb C^NWe study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr?dinger operators, acting on , for arbitrary N ≥ 1. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval , they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.      

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.     

Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents

A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is certtral to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Roessler system.     

Fractal differential equations and fractal-time dynamical systems

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledF ^α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF ^α-integral andF ^α-derivative respectively. TheF ^α-integral is suitable for integrating functions with fractal support of dimension α, while theF ^α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofF ^α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form

On the validity of stochastic rate equations in finite systems with finite-strength interactions

Starting with the Hamiltonian for a linear harmonic chain of 2N particles of massm and one of massM, we have carried out numerical calculations for the momentum autocorrelation function of the mass defect particle for chains with finite numberN of mass points and for nonzero values of the mass ratio=m/M. These results have been compared with the well-known exponential relaxation of the momentum autocorrelation function which is found to be the rigorous result when passing to the thermodynamic and weak-coupling limit. In these limits, the dynamics of the mass defect particle is exactly described by a Fokker-Planck equation, i.e., a stochastic equation of motion. We have shown that, to an excellent approximation, an exponential relaxation of the momentum autocorrelation function is obtained for mass ratios as high as=0.1 and for chains with only 50 particles. Thus, for the harmonic chain considered here, the stochastic equations of motion can be applied to a very good approximation far outside the usually imposed thermodynamic and weak-coupling limits.Supported in part by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018 and by the National Science Foundation under Grant GP28257X.     

Phase Desynchronization as a Mechanism for Transitions to High—Dimensional Chaos

Phase is an important degree of freedom in studies of chaotic oscillations.Phase coherence and localization in coupled chaotic elements are studied.It is shown that phase desynchronization is a key mechanism responsible for the transitions from low-to high-dimensional chaos.The route from low-dimensional chaos to high-dimensional toroidal chaos is accompanied by a cascade of phase desynchronizations.Phase synchronization tree is adopted to exhibit the entrainment process.This bifurcation tree implies an intrinsic cascade of order embedded in irregular motions.     

Effect of noise on chaos synchronization in time-delayed systems: Numerical and experimental observations

We discuss the constructive role of noise (white and colored) in chaos synchronization in time-delayed systems. We first numerically investigate noise-induced synchronization (NIS) between two identical uncoupled Ikeda and Mackey–Glass systems. We find that synchronization occurs above a critical noise intensity that differs for different colors of noise. Synchronization onset is characterized by the value of the maximum transverse Lyapunov exponent. We then discuss the enhancement of chaos synchronization between two time-delayed systems when they are coupled unidirectionally. The effect of parameter mismatch for NIS is described in detail. We provide experimental evidence of NIS for a Mackey–Glass-like system in an electronic circuit using different colors of noise. An integration scheme for time-delayed systems in the presence of additive white and colored noise is discussed.