Birkhoff系统的梯度表示和分数维梯度表示

A Gradient Representation and a Fractional Gradient Representation of Birkhoff System

研究Birkhoff系统的梯度表示和阶α=2的分数维梯度表示.首先给出Birkhoff系统成为通常梯度系统的条件.然后,给出Birkhoff系统成为阶α=2的分数维梯度系统的条件.当梯度系统的势函数V可选为Lyapunov函数时,可用Lyapunov定理来研究系统的稳定性.同时,因为梯度系统的线性化系统的特征方程仅有实根,可按Lyapunov一次近似理论来研究系统的稳定性.最后,举例说明结果的应用.

The gradient representation and the fractional gradient representation with order α = 2 of the Birkhoff system are studied. Firstly, the condition under which the Birkhoff system could be considered as α general gradient system is given. Secondly, the condition under which the system could be considered as a fractional gradient system with order a=2 is also obtained. For a gradient system, one could study its stability by using the Lyapunov theorem when its potential function V can be chosen as a Lyapunov function. Another, since the roots of characteristic equation of linearized system of the gradient system are real, then the stability of the system can be discussed by the Lyapunov＇s first-order approximate theory. Finally, two examples are given to illustrate the application of the results.