Generalized energy integral and reduction of rotational relativistic Birkhoffian system

We study the order reduction method of the rotational relativistic Birkhoffian equations.For a rotational relativistic autonomous Birkhoffian system,if the conservative law of the Birkhoffian holds,the conservative quantity can be called the generalized energy integral.Through the eneralized energy integral,the order of the system can be reduced.If the rotational realtivistic Birkhoffian system has a generalized energy integral,then the Birkhoffian equations can be reduced by at least two degrees and the Birkhoffian form can be kept.An example is given to illustrate the application of the result.     

Whittaker Order Reduction Method of Relativistic Birkhoffian Systems

The order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic autonomous Birkhoffian system, if the conservative law of the Birkhoffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the relativistic Birkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics and the relativistic Lagrangian mechanics are discussed, and the Whittaker order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.     

Whittaker Order Reduction Method of Relativistic Birkhoffian Systems

The order reduction method of the relativistic Birkhollian equations is studied. For a relativistic autonomous Birkhotffian system, if the conservative law of the Birkhotffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the relativisticBirkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least twodegrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics and the relativistic Lagrangian mechanics are discussed, and the Whittaker order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of theresult.     

转动相对论Birkhoff约束系统积分不变量的构造

研究转动相对论Birkhoff约束系统积分不变量的构造首先,建立转动相对论系统的约束Birkhoff方程;其次,利用等时变分与非等时变分之间的关系建立系统的非等时变分方程;然后,研究转动相对论Birkhoff约束系统的第一积分与积分不变量之间的关系,证明由系统的一个第一积分可以构造一个积分不变量,并给出自由Birkhoff系统的相应结果;最后,讨论转动相对论Hamilton系统、相对论Birkhoff系统和Hamilton系统、经典转动系统和等时变分情形下的积分不变量的构造,结果表明相关的结论均为该定理的特款给出一个例子说明结果的应用 关键词： 转动相对论 Birkhoff系统 约束 第一积分 积分不变量     

Phase-coupled nonlinear dynamical systems, stability, first integrals, and Liapunov exponents

The stability of a class of coupled identical autonomous systems of first-order nonlinear ordinary differential equations is investigated. These couplings play a central role in controlling chaotic systems and can be applied in electronic circuits and laser systems. As applications we consider a coupled van der Pol equation and a coupled logistic map. When the uncoupled system admits a first integral we study whether a first integral exists for the coupled system. Gradient systems and the Painlevé property are also discussed. Finally, the relation of the Liapunov exponents of the uncoupled and coupled systems are discussed.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoflan system, the first integrals and the construction of integral invariants are studied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfect differential method. Secondly, the equations of nonsimultaneous variation of the system are established by using the relation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the first integral and the integral invariant of the system is studied, and it is proved that, using a t~rst integral, we can construct an integral invarlant of the system. Finally, the relation between the relativistic Birkhoflan dynamics and the relativistic Hamilton;an dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltonian system are obtained. Two examples are given to illustrate the application of the results.     

自治广义Birkhoff系统的平衡稳定性

研究广义Birkhoff系统的平衡稳定性问题.建立了自治广义Birkhoff系统的平衡方程;给出了自治广义Birkhoff系统的一次近似方程,利用Lyapunov一次近似理论,建立了系统平衡状态稳定性的判据;构建了Lyapunov函数,利用Lyapunov直接法,建立了自治广义Birkhoff系统平衡状态稳定性的判据.给出了若干算例以说明结果的应用.     

Lagrangians and Hamiltonians for One-Dimensional Autonomous Systems

An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system that has certain quasi-relativistic properties. A new method based on a Taylor series expansion is used to obtain the associated Hamiltonian for this system. These results have the usual expression for a conservative system when the dissipation parameter goes to zero. An example of this approach is given.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the first integrals and the construction of integral invariants arestudied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfectdifferential method. Secondly, the equations of nonsimultaneous variation of the system are established by using therelation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the firstintegral and the integral invariant of the system is studied, and it is proved that, using a first integral, we can construct anintegral invariant of the system. Finally, the relation between the relativistic Birkhoffian dynamics and the relativisticHamiltonian dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltoniansystem are obtained. Two examples are given to illustrate the application of the results.     

转动相对论系统动力学的积分理论

建立转动相对论系统动力学方程的积分理论.给出系统运动的第一积分,分别利用系统的循环积分和能量积分降阶运动方程,得到推广的Routh方程和推广的Whittaker方程,建立系统运动的正则方程和变分方程,并由第一积分构造系统的积分不变量.给出系统的Poincaré-Cartan型积分变量关系和积分不变量. 关键词： 转动相对论 运动方程 积分方法     

广义Birkhoff自治系统平衡状态的流形稳定性

研究广义Birkhoff自治系统平衡状态流形稳定性.建立广义Birkhoff自治系统的受绕运动方程和平衡方程.由Liapunov稳定性理论给出广义Birkhoff自治系统的平衡状态流形稳定性的有关判据.举例说明结果的应用.     

A potential integration method for Birkhoffian system

This paper is intended to apply the potential integration method to the differential equations of the Birkhoffian system. The method is that, for a given Birkhoffian system, its differential equations are first rewritten as 2n first-order differential equations. Secondly, the corresponding partial differential equations are obtained by potential integration method and the solution is expressed as a complete integral. Finally, the integral of the system is obtained.     

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.     

求解微分方程的Hojman方法

首先，将Hojman用于求解二阶微分方程组守恒量的方法推广并应用于一阶微分方程组，特别是奇数维微分方程组的积分问题.然后，证明 Hojman定理是本文定理的特殊情形.最后，举例说明结果的应用. 关键词： 微分方程 Hojman定理 守恒量 积分     

广义Birkhoff系统的积分不变量

研究广义Birkhoff系统的积分不变量.给出系统存在积分不变量的条件,在此条件下导出系统的线性积分不变量、通用积分不变量和二阶绝对积分不变量.举例说明结果的应用. 关键词： 广义Birkhoff方程 线性积分不变量 通用积分不变量 二阶绝对积分不变量     

A new four-dimensional chaotic system with first Lyapunov exponent of about 22,hyperbolic curve and circular paraboloid types of equilibria and its switching synchronization by an adaptive global integral sliding mode control

This paper presents a new four-dimensional(4 D) autonomous chaotic system which has first Lyapunov exponent of about 22 and is comparatively larger than many existing three-dimensional(3 D) and 4 D chaotic systems.The proposed system exhibits hyperbolic curve and circular paraboloid types of equilibria.The system has all zero eigenvalues for a particular case of an equilibrium point.The system has various dynamical behaviors like hyperchaotic,chaotic,periodic,and quasi-periodic.The system also exhibits coexistence of attractors.Dynamical behavior of the new system is validated using circuit implementation.Further an interesting switching synchronization phenomenon is proposed for the new chaotic system.An adaptive global integral sliding mode control is designed for the switching synchronization of the proposed system.In the switching synchronization,the synchronization is shown for the switching chaotic,stable,periodic,and hybrid synchronization behaviors.Performance of the controller designed in the paper is compared with an existing controller.     

自主协同系统的宏观稳定机理研究

为保证自主协同机群的内聚性, 实现多机的有效沟通和协作, 借鉴蜂王信息素机理, 抽象出跟踪型多无人机自主协同系统的宏观运动特征. 通过构造Lyapunov函数, 分析该系统的稳定性, 进而得到系统的稳定性判据. 仿真结果表明: 本文所提的稳定控制机理不仅能够保证自主协同系统的稳定性, 还能够通过调整相关控制参数有效地控制系统规模.     

Calculations of periodic orbits: The monodromy method and application to regularized systems

We describe a numerical method for calculating periodic orbits, which is a generalization of the monodromy method by Baranger et al. to the case of an arbitrary autonomous dynamical system. Two variants of the method are developed, using the midpoint and the Runge-Kutta discretization of equations of motion, respectively. Particularly, we adapt the first variant for calculating periodic orbits of Hamiltonian systems when the period or the energy is given a priori. Finally, we consider the application of the monodromy method to the case of regularized mechanical systems and demonstrate the use by two examples. (c) 1999 American Institute of Physics.