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.     

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.     

Fractional conformal invariance method for finding conserved quantities of dynamical systems

For a dynamical system that can be transformed into fractional Birkhoffian representation, under a more general kind of fractional infinitesimal transformation of Lie group, we present the fractional conformal invariance method and it is found that, using the new method, we can find a new kind of non-Noether conserved quantity; and we find that, as a special case, an autonomous fractional Birkhoffian system possesses more conserved quantities. Also, as the fractional conformal invariance method’s applications, we, respectively, explore the conformal invariance and conserved quantities of a fractional Lotka biochemical oscillator and a fractional Hojman–Urrutia model. This work constructs a basic theoretical framework of fractional conformal invariance method, and provides a general method for finding conserved quantities of an actual fractional dynamical system that is related to science and engineering.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

Stochastic fractional optimal control of quasi-integrable Hamiltonian system with fractional derivative damping

A stochastic fractional optimal control strategy for quasi-integrable Hamiltonian systems with fractional derivative damping is proposed. First, equations of the controlled system are reduced to a set of partially averaged It $\hat{o}$ stochastic differential equations for the energy processes by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and a stochastic fractional optimal control problem (FOCP) of the partially averaged system for quasi-integrable Hamiltonian system with fractional derivative damping is formulated. Then the dynamical programming equation for the ergodic control of the partially averaged system is established by using the stochastic dynamical programming principle and solved to yield the fractional optimal control law. Finally, an example is given to illustrate the application and effectiveness of the proposed control design procedure.     

Mei symmetries and conserved quantities for non-conservative Hamiltonian difference systems with irregular lattices

Noether conserved quantities and Mei symmetries for non-conservative Hamiltonian difference systems with irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie point transformations acting on the lattice, as well as the difference equations, and the determining equations of Mei symmetries are obtained for the systems. The discrete versions of Noether conserved quantity are constructed by the Mei symmetries. An example is presented to illustrate the results.     

Noether symmetry method for Birkhoffian systems in terms of generalized fractional operators

Compared with the Hamiltonian mechanics and the Lagrangian mechanics, the Birkhoffian mechanics is more general. The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators, which are proposed recently. Therefore, differential equations of motion within generalized fractional operators are established. Then, in order to find the solutions to the differential equations, Noether symmetry, conserved quantity, perturbation to Noether symmetry and adiabatic invariant are investigated. In the end, two applications are given to illustrate the methods and results.     

NOETHER QUASI-SYMMETRY AND APPROXIMATE NOETHER CONSERVATION LAWS FOR WEAKLY NONLINEAR DYNAMICAL EQUATIONS 1)

自然界和工程技术领域存在大量的非线性问题,它们通常需要用非线性微分方程来描述. 守恒量在微分方程的求解、约化和定性分析方面发挥重要作用. 因此,研究非线性动力学方程的近似守恒量具有重要意义. 文章利用 Noether 对称性方法研究弱非线性动力学方程的近似守恒量. 首先,将弱非线性动力学方程化为一般完整系统的 Lagrange 方程,在 Lagrange 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 其次,将弱非线性动力学方程化为相空间中一般完整系统的 Hamilton 方程,在 Hamilton 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 再次,将弱非线性动力学方程化为广义 Birkhoff 方程,在 Birkhoff 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 最后,以著名的 van der Pol 方程,Duffing 方程以及弱非线性耦合振子为例,分析三个不同框架下弱非线性系统的 Noether 准对称性与近似 Noether 守恒量的计算. 结果表明：同一弱非线性动力学方程可以化为不同的一般完整系统或不同的广义 Birkhoff 系统;Hamilton 框架下的结果是 Birkhoff 框架的特例,而 Lagrange 框架下的结果与 Hamilton 框架的等价. 利用 Noether 对称性方法寻找弱非线性动力学方程的近似守恒量不仅方便有效,而且具有较大的灵活性.     

Birkhoff系统的广义正则变换及其基本形式

动力学方程的积分问题是分析力学研究的一个重要方面．由于求解一般的动力学方程往往会遇到很大困难，因此可利用变量变换，使方程变得容易求解．文章研究Birkhoff系统的广义正则变换．首先，建立Birkhoff系统的广义正则变换的充分必要条件；其次，基于该条件，给出Birkhoff系统的广义正则变换的六种基本形式，导出每一种情况下新旧变量之间的变换关系．作为特例，文中给出Hamilton方程的正则变换．文末，给出算例以说明结果的应用．     

弱非线性动力学方程的 Noether 准对称性与近似 Noether 守恒量

自然界和工程技术领域存在大量的非线性问题,它们通常需要用非线性微分方程来描述. 守恒量在微分方程的求解、约化和定性分析方面发挥重要作用. 因此,研究非线性动力学方程的近似守恒量具有重要意义. 文章利用 Noether 对称性方法研究弱非线性动力学方程的近似守恒量. 首先,将弱非线性动力学方程化为一般完整系统的 Lagrange 方程,在 Lagrange 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 其次,将弱非线性动力学方程化为相空间中一般完整系统的 Hamilton 方程,在 Hamilton 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 再次,将弱非线性动力学方程化为广义 Birkhoff 方程,在 Birkhoff 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 最后,以著名的 van der Pol 方程,Duffing 方程以及弱非线性耦合振子为例,分析三个不同框架下弱非线性系统的 Noether 准对称性与近似 Noether 守恒量的计算. 结果表明:同一弱非线性动力学方程可以化为不同的一般完整系统或不同的广义 Birkhoff 系统;Hamilton 框架下的结果是 Birkhoff 框架的特例,而 Lagrange 框架下的结果与 Hamilton 框架的等价. 利用 Noether 对称性方法寻找弱非线性动力学方程的近似守恒量不仅方便有效,而且具有较大的灵活性.      

Symmetries and Conserved Quantities of Stochastic Dynamical Control Systems

A new definition is given for both exact and quasi symmetries of Itô and Stratonovich dynamical control systems. Determining systems of symmetries for these systems have been obtained and their relation is discussed. It is shown that conserved quantities can be found from both exact and quasi symmetries of stochastic dynamical control systems, which includes Hamiltonian control systems as a special case. Systems which can be controlled via conserved quantities have been investigated. Results have been applied to the control of an N-species stochastic Lotka—Volterra system.     

时间尺度上Hamilton系统的Noether理论

提出并研究时间尺度上Hamilton系统的Noether对称性与守恒量问题．建立了时间尺度上Hamilton原理,导出了相应的Hamilton正则方程．基于时间尺度上Hamilton作用量在群的无限小变换下的不变性,建立了时间尺度上Hamilton系统的Noether定理．定理的证明分成两步：第一步,在时间不变的无限小变换群下给出证明;第二步,利用时间重新参数化技术得到了一般无限小变换群下的定理．给出了经典和离散两种情况下Hamilton系统的Noether守恒量．文末举例说明结果的应用．     

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.     

Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理

应用分数阶模型可以更准确地描述和研究复杂系统的动力学行为和物理过程,同时Birkhoff力学是Hamilton力学的推广,因此研究分数阶Birkhoff系统动力学具有重要意义.分数阶Noether定理揭示了Noether对称变换与分数阶守恒量之间的内在联系,但是当变换拓展为Noether准对称变换时,该定理的推广遇到了很大的困难.本文基于时间重新参数化方法提出并研究Caputo导数下分数阶Birkhoff系统的Noether准对称性与守恒量.首先,将时间重新参数化方法应用于经典Birkhoff系统的Noether准对称性与守恒量研究,建立了相应的Noether定理;其次,基于分数阶Pfaff作用量分别在时间不变的和一般单参数无限小变换群下的不变性给出分数阶Birkhoff系统的Noether准对称变换的定义和判据,基于Frederico和Torres提出的分数阶守恒量定义,利用时间重新参数化方法建立了分数阶Birkhoff系统的Noether定理,从而揭示了分数阶Birkhoff系统的Noether准对称性与分数阶守恒量之间的内在联系.分数阶Birkhoff系统的Noether对称性定理和经典Birkhoff系统的Noether定理是其特例.最后以分数阶Hojman-Urrutia问题为例说明结果的应用.     

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.     

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.     

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate the application of the result.     

Algebraic structures and poisson integrals of relativistic dynamical equations for rotational systems

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Fractional Birkhoffian method for equilibrium stability of dynamical systems

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.     

A numerical method for fractional integral with applications

IntroductionThefractionalcalculushasalonghistoryandthereareamassofworkstodiscussthefractionalderivativesandfractionalintegralswitharbitrary (realorcomplex)order[1- 3 ].Thefractionalcalculushasawideapplicationbackground ,especiallyinthefieldsofchemistry ,electromagnetics,materialscienceandmechanics.Forexample,Gement[4 ]proposedthefractionalderivativeconstitutivemodelsofaviscoelasticmaterialatfirst.Themodelshavereceivedincreasingattention[5 - 7].Onlyafewparametersarecontainedinthemodelsandthemo…