W.Q. Zhu

Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals ofmotion can be constructed in nonresonant case. In a resonant case with a resonant relations, an (n +α)-dimensional averaged FPK equation governing the transition probability density of n action variables and a combinations ofphase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

An Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems

A strategy for optimal nonlinear feedback control of randomlyexcited structural systems is proposed based on the stochastic averagingmethod for quasi-Hamiltonian systems and the stochastic dynamicprogramming principle. A randomly excited structural system isformulated as a quasi-Hamiltonian system and the control forces aredivided into conservative and dissipative parts. The conservative partsare designed to change the integrability and resonance of the associatedHamiltonian system and the energy distribution among the controlledsystem. After the conservative parts are determined, the system responseis reduced to a controlled diffusion process by using the stochasticaveraging method. The dissipative parts of control forces are thenobtained from solving the stochastic dynamic programming equation. Boththe responses of uncontrolled and controlled structural systems can bepredicted analytically. Numerical results for a controlled andstochastically excited Duffing oscillator and a two-degree-of-freedomsystem with linear springs and linear and nonlinear dampings, show thatthe proposed control strategy is very effective and efficient.

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

Abstract An n degree-of-freedom Hamiltonian system with r (1 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 Ito 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 Ito 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.

Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation

Abstract A harmonic function with constant amplitude and random frequency and phase is called bounded noise. In this paper, the effect of bounded noise on the chaotic behavior of the Duffing oscillator under parametric excitation is studied in detail. The random Melnikov process is derived and a mean-square criterion is used to detect the chaotic dynamics in the system. It is found that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the noise in frequency increases. The threshold of bounded noise amplitude for the onset of chaos is also determined by the numerical calculation of the largest Lyapunov exponents. The effect of bounded noise on the Poincare map and power spectra is also investigated. The numerical results qualitatively confirm the conclusion drawn by using the random Melnikov process with mean-square criterion for larger noise intensity.

Lyapunov Exponents and Stochastic Stability of Quasi-Integrable-Hamiltonian Systems

The averaged equations of integrable and non resonant Hamiltonian systems of multi-degree-of-freedom subject to light damping and real noise excitations of small intensities are first derived. Then, the expression for the largest Lyapunov exponent of the square root of the Hamiltonian is formulated by generalizing the well-known procedure due to Khasminskii to the averaged equations, from which the stochastic stability and bifurcation phenomena of the original systems can be determined approximately. Linear and nonlinear stochastic systems of two degrees-of-freedom are investigated to illustrate the application of the proposed combination approach of the stochastic averaging method for quasi-integrable Hamiltonian systems and Khasminskiis procedure.

First-passage failure of quasi-non-integrable-Hamiltonian systems

The first-passage failure of multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems with Gaussian white noise excitations is investigated. Based on the stochastic averaging method for quasi-non-integrable-Hamiltonian systems, the Hamiltonian can be approximated as a one-dimensional diffusion process, from which a backward Kolmogorov equation for conditional reliability function and generalized Pontryagin equations for the moments of first passage time can be established. The conditional reliability function, the conditional probability density of first-passage time and the moments of the first-passage time of any order can be obtained by solving these equations with suitable initial and boundary conditions. Two examples are studied in detail to illustrate the above procedure.

First-passage failure of quasi-integrable Hamiltonian systems

The first-passage failure of quasi-integrable Hamiltonian si-stems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

Nonlinear stochastic optimal control of partially observable linear structures

Abstract A strategy for nonlinear stochastic optimal control of partially observable linear structures is proposed and illustrated with linear building structures equipped with control devices and sensors under horizontal ground acceleration excitation. The control problem of a partially observable structure is first converted into that of a completely observable structure based on the separation principle. Then, a partially averaged control system of Ito equations is obtained from the completely observable structure by using the stochastic averaging method for quasi-Hamiltonian systems. Dynamical programming equations for finite and infinite time-interval controls are established based on the stochastic dynamical programming principle and solved to obtain the optimal control law and value function. Finally, the response of controlled structure is obtained from solving the Fokker–Planck–Kolmogorov equation associated with the fully averaged system of Ito equations. The numerical results for a five-story building structure model are obtained by using the proposed control strategy and compared with those by using linear quadratic Gaussian control strategy to show the effectiveness and efficiency of the proposed strategy.

Stochastic Hopf bifurcation of quasi-nonintegrable-Hamiltonian systems

Abstract A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is first reduced to an one-dimensional Ito stochastic differential equation for the averaged Hamiltonian by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Then the relationship between the qualitative behavior of the stationary probability density of the averaged Hamiltonian and the sample behaviors of the one-dimensional diffusion process of the averaged Hamiltonian near the two boundaries is established. Thus, the stochastic Hopf bifurcation of the original system is determined approximately by examining the sample behaviors of the averaged Hamiltonian near the two boundaries. Two examples are given to illustrate and test the proposed procedure.

Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ≤ n - 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.