Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Hénon–Heiles model, and we study its stability of the manifolds of equilibrium states.     

Stability for manifolds of equilibrium states of generalized Hamiltonian systems

For a generalized Hamiltonian system, stability for the manifolds of equilibrium states is presented based on Lyapunov’s stability theories. Equilibrium equations, perturbation equations and first approximate equations of the system are given. A theorem for the stability of manifolds of equilibrium states of general autonomous system is used to the generalized Hamiltonian system, and three propositions on the stability of manifolds of equilibrium states of the system are obtained. Two examples are given to illustrate application of the method and results.     

Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives

In this paper, we present a new kind of fractional dynamical equations, i.e., the fractional generalized Hamiltonian equations in terms of combined Riesz derivatives, and it is proved that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the Poisson conservation law of the fractional generalized Hamiltonian system is investigated. Then the conditions, which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given. Further, the conserved quantities of a fractional dynamical system are given to illustrate the method and results of the application. At last, a new fractional Volterra model of the three species groups is presented and its conserved quantities are obtained, by using the method of this paper.     

Birkhoff 系统稳定性的动力学控制

本文研究 Birkhoff 系统和广义 Birkhoff 系统平衡稳定性的动力学控制. 首先建立系统的运动方程和平衡方程. 其次,研究 Birkhoff 系统中控制参数出现在 Birkhoff 函数中平衡稳 定性的动力学控制. 方法是通过选取控制参数使得 Birkhoff 函数 $B$ 成为定号函数,而其时间导数 $\dot {B}$ 为与 $B$ 反号的常号函数. 再次,研究广义 Birkhoff 系统平衡稳定性的动力学控制,通过选取 Birkhoff 函数或附加项中包含控制参数的方法,使得 Birkhoff 函数是定号函数,而其时间导数为反号的常号函数,从而控制系统的平衡稳定性. 最后举例说明结果的应用.      

DYNAMICAL CONTROL OF STABILITY FOR BIRKHOFFIAN SYSTEM$^{\bf 1)}$

本文研究 Birkhoff 系统和广义 Birkhoff 系统平衡稳定性的动力学控制. 首先建立系统的运动方程和平衡方程. 其次,研究 Birkhoff 系统中控制参数出现在 Birkhoff 函数中平衡稳 定性的动力学控制. 方法是通过选取控制参数使得 Birkhoff 函数 $B$ 成为定号函数,而其时间导数 $\dot {B}$ 为与 $B$ 反号的常号函数. 再次,研究广义 Birkhoff 系统平衡稳定性的动力学控制,通过选取 Birkhoff 函数或附加项中包含控制参数的方法,使得 Birkhoff 函数是定号函数,而其时间导数为反号的常号函数,从而控制系统的平衡稳定性. 最后举例说明结果的应用.     

Fractional generalized Hamiltonian equations and its integral invariants

In this paper, we present a new kind of fractional dynamical equations, i.e. the fractional generalized Hamiltonian equations, and study variation equations and the method of the construction of integral invariants of the system. Based on the definition of Riemann–Liouville fractional derivatives, fractional generalized Hamiltonian equations and its variation equations are established. Then, the relation between first integral and integral invariant of the system is studied, and it is proved that, using a first integral, we can construct an integral invariant of the system. As deductions of above results, a construction method of integral invariants of a traditional generalized Hamiltonian system are given. Further, one example of fractional generalized Hamiltonian system is given to illustrate the method and results of the application. Finally, we study the first integral and integral invariant of the Euler equation of a rigid body which rotates with respect to a fixed-point.     

The generalized Hamiltonian model for the shafting transient analysis of the hydro turbine generating sets

Traditional rotor dynamics mainly focuses on the steady- state behavior of the rotor and shafting. However, for systems such as hydro turbine generating sets (HTGS) where the control and regulation is frequently applied, the shafting safety and stabilization in transient state is then a key factor. The shafting transient state inevitably involves multiparameter domain, multifield coupling, and coupling dynamics. In this paper, the relative value form of the Lagrange function and its equations have been established by defining the base value system of the shafting. Taking the rotation angle and the angular speed of the shafting as a link, the shafting lateral vibration and generator equations are integrated into the framework of generalized Hamiltonian system. The generalized Hamiltonian control model is thus established. To make the model more general, additional forces of the shafting are taken as the input excitation in proposed model. The control system of the HTGS can be easily connected with the shafting model to form the whole simulation system of the HTGS. It is expected that this study will build a foundation for the coupling dynamics theory using the generalized Hamiltonian theory to investigate coupling dynamic mechanism among the shafting vibration, transient of hydro turbine generating sets, and additional forces of the shafting.     

EQUILIBRIUM STABILITY OF GENERALIZED BIRKHOFF’S AUTONOMOUS SYSTEM

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HAMILTONIAN MECHANICS ON KAHLER MANIFOLDS

Using the mechanical principle, the theory of modern geometry and advanced calculus, Hamiltonian mechanics was generalized to Kahler manifolds, and the Hamiltonian mechanics on Kahler manifolds was established. Then the complex mathematical aspect of Hamiltonian vector field and Hamilton's equations was obtained, and so on.     

Feedback Stabilization of Quasi Nonintegrable Hamiltonian Systems by Using Lyapunov Exponent

A procedure for designing a feedback control to asymptotically stabilize, with probability one, a quasi nonintegrable Hamiltonian system is proposed. First, the motion equations of a system are reduced to a one-dimensional averaged Itô stochastic differential equation for controlled Hamiltonian by using the stochastic averaging method for quasi nonintegrable Hamiltonian systems. Second, a dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is established based on the dynamical programming principle. This equation is then solved to yield the optimal control law. Third, a formula for the Lyapunov exponent of the completely averaged Itô equation is derived by introducing a new norm for the definitions of stochastic stability and Lyapunov exponent in terms of the square root of Hamiltonian. The asymptotic stability with probability one of the originally controlled system is analysed approximately by using the Lyapunov exponent. Finally, the cost function is determined by the requirement of stabilizing the system. Two examples are given to illustrate the application of the proposed procedure and the effectiveness of control on stabilizing the system.     

用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统1）

Birkhoff系统是一类比Hamilton系统更广泛的约束力学系统,可在原子与分子物理,强子物理中找到应用.非定常约束力学系统的稳定性研究是重要而又困难的课题,用构造Lyapunov函数的直接方法来研究稳定性问题有很大难度,其中如何构造Lyapunov函数是永远的开放问题.本文给出一种间接方法,即梯度系统方法.提出一类梯度系统,其矩阵是负定非对称的,这类梯度系统的解可以是稳定的或渐近稳定的.梯度系统特别适合用Lyapunov函数来研究,其中的函数V通常取为Lyapunov函数.列出广义Birkhoff系统的运动方程,广义Birkhoff系统是一类广泛约束力学系统.当其中的附加项取为零时,它成为Birkhoff系统,完整约束系统和非完整约束系统都可纳入该系统.给出广义Birkhoff系统的解可以是稳定的或渐近稳定的条件,进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的广义Birkhoff系统.该方法也适合其他约束力学系统.最后用算例说明结果的应用.     

First passage failure of quasi-partial integrable generalized Hamiltonian systems

The first passage failure of quasi-partial integrable generalized Hamiltonian systems is studied by using the stochastic averaging method. First, the stochastic averaging method for quasi-partial integrable generalized Hamiltonian systems is introduced briefly. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are derived from the averaged Itô equations. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions, respectively. Finally, one example is given to illustrate the proposed procedure in detail and the solutions are confirmed by using the results from Monte Carlo simulation of the original system.     

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.     

The activation method for discretized conservative nonlinear stability problems with multiple parameter and state variables

For nonlinear stability problems of discretized conservative systems with multiple parameter variables and multiple state variables,the activation method is put forward,by which activated potential functions and activated equilibrium equations are derived.The activation method is the improvement and enhancement of Liapunov-Schmidt method in elastic stability theory.It is more generalized and more normalized than conventional perturbation methods.The activated potential functions may be transformed into normalized catastrophe potential functions.The activated equilibrium equations may be treated as bifurcation equations.The researches in this paper will motivate the combination of elastic stability theory with catastrophe theory and bifurcation theory.     

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.     

Global dynamical behavior of a three-body system with flexible connection in the gravitational field

Summary In this paper, the global behavior of relative equilibrium states of a three-body satellite with flexible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. Received 10 March 1998; accepted for publication 21 July 1998     

Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance

In this work we study the existence of Silnikov homoclinicorbits in the averaged equations representing the modal interactionsbetween two modes with zero-to-one internal resonance. The fast mode isparametrically excited near its resonance frequency by a periodicforcing. The slow mode is coupled to the fast mode when the amplitude ofthe fast mode reaches a critical value so that the equilibrium of theslow mode loses stability. Using the analytical solutions of anunperturbed integrable Hamiltonian system, we evaluate a generalizedMelnikov function which measures the separation of the stable and theunstable manifolds of an annulus containing the resonance band of thefast mode. This Melnikov function is used together with the informationof the resonances of the fast mode to predict the region of physicalparameters for the existence of Silnikov homoclinic orbits.     

Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions

This paper presents a new method for the stress singularity analysis near the crack corners of a multi-material junctions. The stress singularities near the crack corners of multi-dissimilar isotropic elastic material junctions are studied analytically in terms of the methods developed in Hamiltonian system. The governing equations of plane elasticity in a sectorial domain are derived in Hamiltonian form via variable substitution and variational principle respectively. Both of the methods of global state variable separation and symplectic eigenfunction expansion are used to find the analytical solution of the problem. The relationships among the state vectors in different material spaces are obtained by means of coordinate transformation and consistent conditions between the two adjacent domains. The expression of the original problem is thus changed into a new form where the solutions of symplectic generalized eigenvalues and eigenvectors are needed. The closed form of expressions is established for the stress singularity analysis near the corner with arbitrary vertex angles. Numerical results are presented with several chosen angles and multi-material constants. To show the potential of the new method proposed, a semi-analytical finite element is furthermore developed for the numerical analysis of crack problems.     

First passage failure of quasi non-integrable generalized Hamiltonian systems

The first passage failure of quasi non-integrable generalized Hamiltonian systems is studied. First, the generalized Hamiltonian systems are reviewed briefly. Then, the stochastic averaging method for quasi non-integrable generalized Hamiltonian systems is applied to obtain averaged Itô stochastic differential equations, from which the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of the first passage time are established. The conditional reliability function and the conditional mean of first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, an example of power system under Gaussian white noise excitation is worked out in detail and the analytical results are confirmed by using Monte Carlo simulation of original system.     

A method for the direct evaluation of buckling loads of an imperfect two-bar frame

Summary The pre-critical, critical, and post-critical nonlinear response of an imperfect due to loading eccentricity two-bar frame is thoroughly discussed. In seeking the maximum load-carrying capacity of this non-sway frame, it was qualitatively established that its loss of stability occurs through a limit point and hence, the case of an asymmetric bifurcation can be considered only in an asymptotic sense. After deriving the nonlinear equilibrium equations with unknowns for the two bar axial forces, we can consider such a continuous system as a two-degree-of-freedom model with generalized coordinates the above axial forces. Then, the equilibrium equations and the stability determinant of the frame can be determined in terms of the first and second derivatives of its total potential energy (TPE) with respect to the axial forces. The vanishing of the second variation of the TPE together with the equilibrium equations allows a simple and direct evaluation of the buckling load. Numerical examples demonstrate the efficiency and the reliability of the proposed method. The authors are indebted to the National Technical University of Athens for the financial support of the project “Thalis Grant”, whose partial results are reported in this paper.