Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.     

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker-Planck-Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.     

On parametric asymptotic stability of large-scale systems

The domain of parameter values in which an autonomous large-scale system is uniformly asymptotically stable is estimated. The comparison method with a vector Lyapunov function is chosen for analysis __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 104–114, May 2008.     

轴向变速运动正交各向异性板的动态稳定性

研究面内载荷作用下轴向变速运动正交各向异性薄板的横向振动及其稳定性。利用Galerkin法与平均法,在激励频率为2倍固有频率或为两阶固有频率之和附近时得到自治的常微分方程组;在参数激励频率和激励振幅平面上,分析由共振引发的失稳区域。数值算例验证了面内载荷、轴向速度、加速度参数对失稳区域的影响。     

Stochastic Stabilization of Quasi-Partially Integrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptoticallystabilize, with probability one, a quasi-partially integrableHamiltonian system is proposed. First, the averaged stochasticdifferential equations for controlled r first integrals are derived fromthe equations of motion of a given system by using the stochasticaveraging method for quasi-partially integrable Hamiltonian systems.Second, a dynamical programming equation for the ergodic control problemof the averaged system with undetermined cost function is establishedbased on the dynamical programming principle. The optimal control law isderived from minimizing the dynamical programming equation with respectto control. Third, the asymptotic stability with probability one of theoptimally controlled system is analyzed by evaluating the maximalLyapunov exponent of the completely averaged Itô equations for the rfirst integrals. Finally, the cost function and optimal control forces aredetermined by the requirements of stabilizing the system. An example isworked out in detail to illustrate the application of the proposedprocedure and the effect of optimal control on the stability of thesystem.     

A new approach to almost-sure asymptotic stability of stochastic systems of higher dimension

The almost sure asymptotic stability of higher-dimensional linear stochastic systems and of a special class of nonlinear stochastic systems with homogeneous drift and diffusion coefficients of order one is studied. Based on the well-known Khasminskii's theorem, a new approach for obtaining the regions of almost sure asymptotic stability and instability without evaluating the stationary probability density of the diffusion process defined on unit hypersphere is proposed. Two examples of two and three degree-of-freedom linear stochastic systems are given to illustrate the application and effectiveness of the proposed approach combined with stochastic averaging.     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.     

Technical stability of stationary automatic control systems of variable structure with switched filters

Sufficient conditions for the technical stability in measure of a nonstationary control system with variable structure are established. The controller of the system has feedback-switched filters functioning together with shaper and actuator. It is assumed that the nonstationary parameters of the system vary within given ranges, at a finite rate, with appropriate control laws, with adjustment against mismatch signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach for analysis of technical stability does not involve sliding mode conditions. Criteria of technical instability in measure for the control system under consideration are formulated using the properties of systems of comparison from below. The general criteria of technical stability and instability are applied to nonstationary filtered-control systems of variable structure of the third order. The comparison method based on normalized Lyapunov functions is used __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 110–127, June 2006.     

Global parametric quadratic stabilizability of nonlinear systems with uncertainty

Sufficient conditions for the global PQ-stabilizability of a nonlinear system with uncertainty are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 126–133, June 2008.     

Stochastic stability of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for calculating the largest Lyapunov exponent and determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. The averaged stochastic differential equations (SDEs) of quasi-integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method for quasi-Hamiltonian systems and the stochastic jump-diffusion chain rule. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii's procedure to the averaged SDEs and the stochastic stability of the original systems is determined approximately. An example is given to illustrate the application of the proposed procedure and its effectiveness is verified by comparing with the results from Monte Carlo simulation.     

实噪声参激Hopf分叉系统研究

采用随机平均法、扩散过程的奇点理论、不变测度方法分析了实噪声参激的分叉系统．明确了噪声的影响将使系统出现与原分叉点不同的噪声导致的分叉点,并使分叉类型产生了根本的改变     

Invariant Measures and Lyapunov Exponents for Stochastic Mathieu System

The principal resonance of the stochastic Mathieu oscillator to randomparametric excitation is investigated. The method of multiple scales isused to determine the equations of modulation of amplitude and phase.The behavior, stability and bifurcation of steady state response arestudied by means of qualitative analyses. The effects of damping,detuning, bandwidth, and magnitudes of random excitation are analyzed.The explicit asymptotic formulas for the maximum Lyapunov exponent areobtained. The almost-sure stability or instability of the stochasticMathieu system depends on the sign of the maximum Lyapunov exponent.     

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters is studied.By constructing the LyapunovKrasovskii functional and employing the decomposition technique of interval matrix and Ito's formula,the delay-dependent criteria for the p-moment exponential robust stability are obtained.Numerical examples show the validity and practicality of the presented criteria.     

Global transient dynamics of nonlinear parametrically excited systems

In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.     

On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous right- hand side(with solutions in Filippov's sense)via locally Lipschitz continuous and regular vector Lyapunov functions.A new type of\"set-valued derivative\"of vector Lyapunov functions is introduced,some generalized comparison principles on discontinuous systems are shown.Furthermore,Lyapunov stability theory is developed for a class of discontinu- ous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.     

Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base

The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation.     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

轴向变速粘弹性Rayleigh梁参数共振稳定性

运用近似解析方法和数值方法研究轴向变速运动黏弹性Rayleigh梁的次谐波共振和组合共振的稳定性区域。基于变分原理,考虑梁断面旋转惯性的影响,推导轴向速度有周期波动的微变形梁横向振动的数学模型;采用多尺度方法建立前两阶次谐波共振和组合共振范围内的参数振动的可解性条件;进而确定梁两端简支边界条件下,因共振而产生的失稳区域;通过微分求积方法求解表征细长Rayleigh梁横向振动的运动微分方程。数值算例分析了黏弹性系数和扭转系数对梁振动失稳区域的影响,将数值仿真结果与近似解析方法的结论进行比较。算例表明:近似解析解的精度较高,第一、第二阶主共振的最大误差分别为3.206%、4.213%。