Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

1996 ◽  
Vol 63 (2) ◽  
pp. 493-500 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Q. Yang
Keyword(s):  
Hamiltonian System ◽  
Stationary Solution ◽  
Hamiltonian Systems ◽  
Stochastic Systems ◽  
Integrable Hamiltonian System ◽  
Stationary Solutions ◽  
Integrable Hamiltonian Systems ◽  
Exact Stationary Solution ◽  
Special Cases ◽  
Action Variables

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.

COMPLETELY AND PARTIALLY INTEGRABLE HAMILTONIAN SYSTEMS IN THE NONCOMPACT CASE

2004 ◽  
Vol 01 (03) ◽  
pp. 167-183 ◽  
Author(s):  
EMANUELE FIORANI
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian System ◽  
Time Dependent ◽  
Integrable Hamiltonian Systems ◽  
Invariant Submanifold ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian System

Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.

Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

1997 ◽  
Vol 64 (4) ◽  
pp. 975-984 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang ◽  
Y. Q. Yang
Keyword(s):  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Transition Probability Density ◽  
Special Cases ◽  
Fpk Equation ◽  
Action Variables

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 of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase 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.

scholarly journals Reidemeister torsion and integrable Hamiltonian systems

1999 ◽  
Vol 22 (4) ◽  
pp. 689-704
Author(s):  
Alexander Fel'shtyn ◽  
Hector Sánchez-Morgado
Keyword(s):  
Hamiltonian System ◽  
Spectral Sequence ◽  
Hamiltonian Systems ◽  
Symplectic Manifold ◽  
Integrable Hamiltonian System ◽  
Integrable Hamiltonian Systems ◽  
Reidemeister Torsion ◽  
Isoenergetic Surface

In this paper, we compute the Reidemeister torsion of an isoenergetic surface for the integrable Hamiltonian system on the 4-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Floer-Witten ideas we bring into play the orbits connecting the critical submanifolds.

scholarly journals Stochastic Minimax Vibration Control for Uncertain Nonlinear Quasi-Hamiltonian Systems with Noisy Observations

2019 ◽  
Vol 24 (4) ◽  
pp. 707-716
Author(s):  
Zu-guang Ying ◽  
Rong-chun Hu ◽  
Rong-hua Huan
Keyword(s):  
Kalman Filter ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Extended Kalman Filter ◽  
Control Strategy ◽  
Stochastic Systems ◽  
Uncertain Parameters ◽  
Dynamical Programming ◽  
Worst Case ◽  
Minimax Control

A stochastic minimax control strategy for uncertain nonlinear quasi-Hamiltonian systems with noisy observations under random excitations is proposed based on the extended Kalman filter and minimax stochastic dynamical programming principle. A structure system with smart sensors and actuators is modeled as a controlled, excited and dissipative Hamiltonian system with noisy observations. The differential equations for the uncertain nonlinear quasi-Hamiltonian system with control and observation under random excitation are given first. The estimated nonlinear stochastic control system with uncertain parameters is obtained from the uncertain quasi-Hamiltonian system with noisy observation. In this case, the optimally estimated state is determined by the observation based on the extended Kalman filter. The dual dynamical programming equation for the estimated uncertain system is then obtained based on the minimax stochastic dynamical programming principle. The worst-case disturbances are determined for bounded uncertain parameters and the optimal control law is determined for the worst case by the programming equation. The proposed minimax control strategy is applied to two uncertain nonlinear stochastic systems with controls and noisy observations. The control effectiveness for the stochastic vibration response reductions of the systems is illustrated with numerical results. The proposed minimax control strategy is applicable to general uncertain nonlinear multi-degree-of-freedom structure systems with noisy observations under random excitations.

Dynamics Around Minimal Hyperbolic Torus in Hamiltonian Systems

2003 ◽  
Vol 17 (22n24) ◽  
pp. 3950-3963
Author(s):  
Jun Yan ◽  
Chong-Qing Cheng
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian System ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Minimal Sets ◽  
System A

By investigating the topological properties of action minimal sets near a minimal hyperbolic torus in nearly integrable Hamiltonian system, a sufficient condition of existence of connections between different Mather sets is obtained.

Feedback Stabilization of Quasi-Integrable Hamiltonian Systems

2003 ◽  
Vol 70 (1) ◽  
pp. 129-136 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang
Keyword(s):  
Feedback Control ◽  
Cost Function ◽  
Hamiltonian Systems ◽  
Infinite Horizon ◽  
Integrable Hamiltonian System ◽  
First Integrals ◽  
Feedback Stabilization ◽  
Largest Lyapunov Exponent ◽  
Dynamical Programming ◽  
Integrable Hamiltonian Systems

A procedure for designing a feedback control to asymptotically stabilize with probability one quasi-integrable Hamiltonian system is proposed. First, a set of averaged Ito^ stochastic differential equations for controlled first integrals is derived from given equations of motion of the system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, a dynamical programming equation for infinite horizon performance index with unknown cost function is established based on the stochastic dynamical programming principle. Third, the asymptotic stability with probability one of the optimally controlled system is analyzed by evaluating the largest Lyapunov exponent of the fully averaged Ito^ equations for the first integrals. Finally, the cost function and feedback control law are determined by the requirement of stabilization of the system. An example is worked out in detail to illustrate the application of the proposed procedure and the effect of optimal control on the stability of the system.

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

1999 ◽  
Vol 66 (1) ◽  
pp. 211-217 ◽  
Author(s):  
W. Q. Zhu ◽  
Z. L. Huang
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Stability ◽  
Degrees Of Freedom ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Largest Lyapunov Exponent ◽  
Integrable Hamiltonian Systems ◽  
Averaged Equations ◽  
Bifurcation Phenomena

The averaged equations of integrable and nonresonant 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 Khasminskii’s procedure.

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