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.     

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.     

Maximal Lyapunov exponent and almost-sure sample stability for second-order linear stochastic system

The principal resonance of second-order system to random parametric excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The effects of damping, detuning, bandwidth, and magnitudes of random excitation are analyzed. The explicit asymptotic formulas for the maximum Lyapunov exponent is obtained. The almost-sure stability or instability of the stochastic Mathieu system depends on the sign of the maximum Lyapunov exponent.     

Optimal bounded control for minimizing the response of quasi-integrable Hamiltonian systems

A procedure for designing optimal bounded control to minimize the response of quasi-integrable Hamiltonian systems is proposed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The equations of motion of a controlled quasi-integrable Hamiltonian system are first reduced to a set of partially completed averaged Itô stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the dynamical programming equation for the control problems of minimizing the response of the averaged system is formulated based on the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints without solving the dynamical programming equation. The response of optimally controlled systems is predicted through solving the Fokker-Planck-Kolmogrov equation associated with fully completed averaged Itô equations. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.     

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.     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

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.     

First-passage failure and its feedback minimization 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. 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 procedures for studying the first-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging method for quasi-partially integrable Hamiltonian systems is briefly reviewed. Then, based on the averaged Itô equations, a backward Kolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of first-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and of maximization of mean first-passage time are formulated. The relationship between the backward Kolmogorov equation and the dynamical programming equation for reliability maximization, and that between the Pontryagin equation and the dynamical programming equation for maximization of mean first-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the effectiveness of feedback control in reducing first-passage failure.     

Stochastic stabilization of quasi non-integrable Hamiltonian systems

A procedure for designing a feedback control to asymptotically stabilize in probability a quasi non-integrable Hamiltonion system is proposed. First, an one-dimensional averaged Itô stochastic differential equation for controlled Hamiltonian is derived from given equations of motion of the system by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Second, a dynamical programming equation for an ergodic control problem with undetermined cost function is established based on the stochastic dynamical programming principle and solved to yield the optimal control law. Third, the asymptotic stability in probability of the system is analysed by examining the sample behaviors of the completely averaged Itô differential equation at its two boundaries. Finally, the cost function and the optimal control forces are determined by the requirement of stabilizing the system. Two examples are given to illustrate the application of the proposed procedure and the effect of control on the stability of the system.     

Moment Lyapunov exponent of three-dimensional system under bounded noise excitation

In the present paper,the moment Lyapunov exponent of a codimensional two-bifurcation system is evaluted,which is on a three-dimensional central manifold and subjected to a parametric excitation by the ...     

Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged Itô stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged Itô equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged Itô equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.     

On the stability of systems with stochastically variable parameters

The paper analyzes questions related to the construction of dynamic stability boundaries of elastic systems subjected to stochastic parametric excitation. It is supposed that the parametric action is a combination of a deterministic static component and a stochastic fluctuating component. The fluctuating component is taken to be a stationary ergodic process. The stability boundaries are built in the region of combination resonance using the stochastic averaging method and a probabilistic approach due to Khasminskii. In this connection, the stochastic averaging method based on the Stratonovich-Khasminskii theory is used. The probabilistic approach consists in using explicit asymptotic expressions for the largest Lyapunov exponent, from which the asymptotic stability boundaries are determined. As an application, the stability of a simply supported thin-walled bar subjected to a stochastically varying longitudinal load is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 128–138, December 2005.     

Stochastic stability of non-linear SDOF systems

A semi-analytical procedure for obtaining stability conditions for strongly non-linear single degree of freedom system (SDOF) subjected to random excitations is presented using stochastic averaging technique. The method is useful for finding stability conditions for systems having highly irregular non-linear functions which cannot be integrated in closed form to yield analytical expressions for averaged drift and diffusion coefficients. In spite of numerical methods available for finding stability of SDOF system by determining Lyapunov exponent, the proposed technique may have to be adopted (i) when the excitation is non-white; and (ii) when numerical integration fails due to convergence problem. The method is developed in such a way that it lends itself to a numerical computational scheme using FFT for obtaining numerical values of drift and diffusion coefficients of Its differential equation and the corresponding FPK equation for the system. These values of averaged drift and diffusion coefficients are then fit into polynomial form using curve fitting technique so that polynomials can be used for stability analysis. Two example problems are solved as illustrations. The first one is the Van der Pol oscillator having non-linearities which can be treated purely analytically. The example is considered for the validation of the proposed method. The second one involves non-linearities in the form of signum function for which purely analytical solution is not possible. The results of the study show that the proposed method is useful and efficient for performing stability analysis of dynamic systems having any type of non-linearities.     

First-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations

In this paper, first-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations is studied theoretically. By using stochastic averaging method, the equations of motion of the original internally resonant Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established under appropriate boundary and (or) initial conditions. An example is given to show the accuracy of the theoretical method. Numerical solutions of high-dimensional backward Kolmogorov and Pontryagin equation are obtained by finite difference. All theoretical results are verified by Monte Carlo simulation.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

On stationary probability density and maximal Lyapunov exponent of a co-dimension two bifurcation system subjected to parametric excitation by real noise

In this paper, the asymptotic expansions of the maximal Lyapunov exponents for a co-dimension two-bifurcation system which is on a three-dimensional center manifold and is excited parametrically by an ergodic real noise are evaluated. The real noise is an integrable function of an n-dimensional Ornstein-Uhlenbeck process. Based on a perturbation method, we examine almost all possible singular boundaries that exist in one-dimensional phase diffusion process. The comparisons between the analytical solutions and the numerical simulations are given. In addition, we also investigate the P-bifurcation behavior for the one-dimensional phase diffusion process. The result in this paper is a further extension of the work by Liew and Liu [1].     

The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.     

The matric algorithm of Lyapunov exponent for the experimental data obtained in dynamic analysis

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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.