Moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation

The moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation are studied in this paper. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter.     

Moment Lyapunov exponents of a two-dimensional system under both harmonic and white noise parametric excitations

The moment Lyapunov exponents and the Lyapunov exponents of a 2D system under both harmonic and white noise excitations are studied. The moment Lyapunov exponents and the Lyapunov exponents are important characteristics determining the moment and almost-sure stability of a stochastic dynamical system. The eigenvalue problem governing the moment Lyapunov exponent is established. A singular perturbation method is applied to solve the eigenvalue problem to obtain second-order, weak noise expansions of the moment Lyapunov exponents. The influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

Stochastic stability of a fractional viscoelastic column under bounded noise excitation

The stability of a viscoelastic column under the excitation of stochastic axial compressive load is investigated in this paper. The material of the column is modeled using a fractional Kelvin–Voigt constitutive relation, which leads to that the equation of motion is governed by a stochastic fractional equation with parametric excitation. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The method of stochastic averaging is used to approximate the responses of the original dynamical system by a new set of averaged variables which are diffusive Markov vector. An eigenvalue problem is formulated from the averaged equations, from which the moment Lyapunov exponent is determined for the column system with small damping and weak excitation. The effects of various parameters on the stochastic stability and significant parametric resonance are discussed and confirmed by simulation results.     

LARGEST LYAPUNOV EXPONENT AND ALMOST CERTAIN STABILITY ANALYSIS OF SLENDER BEAMS UNDER A LARGE LINEAR MOTION OF BASEMENT SUBJECT TO NARROWBAND PARAMETRIC EXCITATION

The first order approximate solutions of a set of non-liner differential equations, which is established by using Kane's method and governs the planar motion of beams under a large linear motion of basement, are systematically derived via the method of multiple scales. The non-linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken into consideration, are analyzed in detail. The largest Lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system.     

Chaotic motion of Van der Pol–Mathieu–Duffing system under bounded noise parametric excitation

The chaotic behavior of Van der Pol–Mathieu–Duffing oscillator under bounded noise is investigated. By using random Melnikov technique, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in this system increases as the intensity of the noise in frequency increases, which is further verified by the maximal Lyapunov exponents of the system. The effect of bounded noise on Poincaré map is also investigated, in addition the numerical simulation of the maximal Lyapunov exponents.     

The largest Lyapunov exponent of chaotic dynamical system in scale space and its application

The largest Lyapunov exponent is an important invariant of detecting and characterizing chaos produced from a dynamical system. We have found analytically that the largest Lyapunov exponent of the small-scale wavelet transform modulus of a dynamical system is the same as the system's largest Lyapunov exponent, both discrete map and continuous chaotic attractor with one or two positive Lyapunov exponents. This property has been used to estimate the largest Lyapunov exponent of chaotic time series with several kinds of strong additive noise.     

Higher-order stochastic averaging to study stability of a fractional viscoelastic column

Stochastic stability of a fractional viscoelastic column axially loaded by a wideband random force is investigated by using the method of higher-order stochastic averaging. By modelling the wideband random excitation as Gaussian white noise and real noise and assuming the viscoelastic material to follow the fractional Kelvin–Voigt constitutive relation, the motion of the column is governed by a fractional stochastic differential equation, which is justifiably and uniformly approximated by an averaged system of Itô stochastic differential equations. Analytical expressions are obtained for the moment Lyapunov exponent and the Lyapunov exponent of the fractional system with small damping and weak random fluctuation. The effects of various parameters on the stochastic stability of the system are discussed.     

Experimental bifurcation analysis of an impact oscillator—Determining stability

We propose and investigate three different methods for assessing stability of dynamical equilibrium states during experimental bifurcation analysis, using a control-based continuation method. The idea is to modify or turn off the control at an equilibrium state and study the resulting behavior. As a proof of concept the three methods are successfully implemented and tested for a harmonically forced impact oscillator with a hardening spring nonlinearity, and controlled by electromagnetic actuators. We show that under certain conditions it is possible to quantify the instability in terms of finite-time Lyapunov exponents. As a special case we study an isolated branch in the bifurcation diagram brought into existence by a 1:3 subharmonic resonance. On this isola it is only possible to determine stability using one of the three methods, which is due to the fact that only this method guarantees that the equilibrium state can be restored after measuring stability.     

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.     

Lyapunov exponents in unstable systems

We investigate the dynamical behavior of unstable systems in the vicinity of the critical point associated with a liquid-gas phase transition. By considering a mean-field treatment, we first perform a linear analysis and discuss the instability growth times. Then, coming to complete Vlasov simulations, we investigate the role of nonlinear effects and calculate the Lyapunov exponents. As a main result, we find that near the critical point, the Lyapunov exponents exhibit a power-law behavior, with a critical exponent beta=0.5. This suggests that in thermodynamical systems the Lyapunov exponent behaves as an order parameter to signal the transition from the liquid to the gas phase.     

The Plateau and Symmetrical Structure of Lyapunov Exponents in 2-Dimensional Homogeneous. Dissipative Map

The universal crossover behavior of Lyapunov exponents in transition from conservative limit to dissipative limit of dynamical system is studied. We discover numerically and prove analytically that for homogeneous dissipative two-dimensional maps, along the equal dissipation line in parameter space, two Lyapunov exponents λ₁ and λ₂ of periodic orbits possess a plateau structure, and around this exponent plateau value, there is a strict symmetrical relation between λ₁ and λ₂ no matter whether the orbit is periodic, quasiperiodic, or chaotic.The method calculating stable window and Lyapunov exponent plateau widths is given. For Hénon map and 2-dimensional circle map, the analytical and numerical results of plateau structure of Lyapunov exponents for period-1,2 and 3 orbits are presented.     

Cycles homoclinic to chaotic sets; robustness and resonance

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient 'bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called 'cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821-823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rossler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. (c) 1997 American Institute of Physics.     

舰船辐射噪声超混沌现象研究

水下弱目标探测和识别一直是水声信号处理领域中研究的难点。从Lyapunov指数谱、吸引子相空间轨迹的演化、分形维数等方面，对船舶辐射噪声是否存在超混沌现象进行了研究。实验结果表明，船舶辐射噪声信号确实存在至少两个正的Lyapunov指数，即存在超混沌现象。辐射噪声吸引子在相空间中的轨迹具有多方向伸展的趋势，且不同类型目标的吸引子具有不同的分形维数。研究结果为建立精确描述辐射噪声信号的非线性模型、为水下弱目标信号探测和识别提供一定的理论依据。     

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.     

Dynamic stability of a slender beam under horizontal–vertical excitations

The dynamic stability of a vertically standing cantilevered beam simultaneously excited in both horizontal and vertical directions at its base is studied theoretically. The beam is assumed to be an inextensible Euler–Bernoulli beam. The governing equation of motion is derived using Hamilton's principle and has a nonlinear elastic term and a nonlinear inertia term. A forced horizontal external term is added to the parametrically excited system. Applying Galerkin's method for the first bending mode, the forced Mathieu–Duffing equation is derived. The frequency response is obtained by the harmonic balance method, and its stability is investigated using the phase plane method. Excitation frequencies in the horizontal and vertical directions are taken as 1:2, from which we can investigate the influence of the forced response under horizontal excitation on the parametric instability region under vertical excitation. Three criteria for the instability boundary are proposed. The influences of nonlinearities and damping of the beam on the frequency response and parametric instability region are also investigated. The present analytical results for instability boundaries are compared with those of experiments carried out by one of the authors.     

Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems

Many flows in nature are “open flows” (e.g. pipe flow). We study two open-flow systems driven by low-level external noise: the time-dependent generalized Ginzburg-Landau equation and a system of coupled logistic maps. We find that a flow which gives every appearance of being chaotic may nonetheless have no positive Lyapunov exponents. By generalizing the notions of convective instability and Lyapunov exponents we define a measure of chaos for these flows.     

基于Lyapunov指数的摇摆条件下自然循环流动不稳定性混沌预测

运用基于最大Lyapunov指数的混沌预测方法对摇摆条件下自然循环系统的流量脉动进行了预测. 对不规则复合型脉动的流量脉动实验数据进行相空间重构, 计算关联维数、二阶Kolmogorov熵和最大Lyapunov指数等几何不变量, 在说明不规则复合型脉动是混沌运动的基础上, 根据最大Lyapunov指数对不规则复合型脉动进行了预测. 通过预测结果和实验结果对比发现: 对于复杂的两相自然循环流动不稳定性, 预测结果具有较高的精度, 说明预测方法的可行性. 同时, 确定了混沌系统可预测的尺度, 提出用动态预测的方式监测系统流量脉动. 本文的研究方法为两相流复杂的流动不稳定性研究提供了新的思路. 关键词： 混沌时间序列 实时预测 最大Lyapunov指数 两相流动不稳定性     

最近邻耦合网络的时空混沌同步研究

本文进行了最近邻网络的时空混沌同步研究.以时空混沌系统作为网络的节点,基于Lyapunov稳定性定理,通过确定网络的最大Lyapunov指数,得到了实现网络完全同步的条件.采用Fisher-Kolmogorov时空混沌系统作为网络节点实例进行了仿真模拟,获得了理想的同步效果.进一步研究了有界噪声影响下网络的同步性能,结果显示它具有较强的抗干扰能力.     

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.