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 state of generalized Hamiltonian system with additional terms

For a generalized Hamiltonian system with additional terms, stability for the manifolds of the equilibrium state is presented. Equilibrium equations, disturbance equations and the first approximate equations of the system are given. A theorem for the stability of the manifolds of the equilibrium state of a general autonomous system is used for the generalized Hamiltonian systems with additional terms, and three propositions on the stability of the manifolds of the equilibrium state of the system are obtained. An example is given to illustrate the application of the method and results. At last, we study the stability for manifolds of the equilibrium state of the Euler equations of a rigid body subjected to external moments of force, by using of the method in this paper.     

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.     

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.     

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.     

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.      

STABILITY FOR THE EQUILIBRIUM STATE OF CHAPLYGIN’S SYSTEMS

The stability for the equitibrium states of Chaplygin’s systems is considered. Theequations of motion of Chaplygin’s systems and the existence conditions of their equilibrium states are given. Some criteria of stability for the equilibrium.states of Chaplygin’s systems are obtained. Two examples are finally given.     

EQUILIBRIUM STABILITY OF GENERALIZED BIRKHOFF’S AUTONOMOUS SYSTEM

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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.     

Nekhoroshev and KAM Stabilities in Generalized Hamiltonian Systems

We present some Nekhoroshev stability results for nearly integrable, generalized Hamiltonian systems, which can be odd dimensional and admit a distinct number of action and angle variables. Using a simultaneous approximation technique due to Lochak, Nekhoroshev stabilities are shown for various cases of quasi-convex generalized Hamiltonian systems along with concrete estimates on stability exponents. Discussions on KAM metric stability of generalized Hamiltonian systems are also given.     

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.     

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.     

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.     

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.     

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.     

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

In this paper,we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous(SD) oscillator with dry friction.The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both( f_n,f_s) and(x,˙x) planes.In the stability analysis,Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and La Salle's invariance principle is employed to obtain the stability of the nonhyperbolic type.Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property,and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property.Numerical calculations show a good agreement with the theoretical analysis and an excellent efficien y of the approach for equilibrium states in this particular Filippov system.Furthermore,the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.     

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.     

Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解

采用辛算法研究了Hamilton体系下介电弹性体圆形薄膜的动力学响应。首先，将该问题引入Hamilton对偶变量体系，借助Legendre变换，给出系统的广义动量和Hamilton函数，通过对Hamilton函数作用量的变分，得到Hamilton体系下的正则方程。其次，对于得到的正则方程给出了辛Runge-Kutta的计算格式。最后，采用二级四阶辛Runge-Kutta算法对动力学系统进行了数值求解，和四级四阶经典Runge-Kutta算法进行对比，结果表明，二级四阶辛Runge-Kutta算法具有保能量以及长时间数值稳定的优势，同时说明四级四阶经典Runge-Kutta算法对于步长依赖的局限性。     

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.     

判定非自治Birkhoff系统稳定性的广义组合梯度方法

研究判定非自治Birkhoff系统稳定性的广义组合梯度方法．首先,给出非自治Birkhoff系统和非自治广义Birkhoff系统的运动微分方程;其次,给出一类将广义梯度系统和广义斜梯度系统组合而成的广义组合梯度系统,并讨论广义组合梯度系统的一些性质;最后,将非自治Birkhoff系统和非自治广义Birkhoff系统在一定条件下表示成广义组合梯度系统,并用广义组合梯度系统的性质研究了这两类Birkhoff系统的稳定性．举例说明结果的应用．