求解非线性系统周期轨道及其周期的一种方法

METHOD FOR DETERMINING PERIODIC ORBIT AND PERIODIC VALUE OF NONLINEAR DYNAMICAL SYSTEM

首先通过时间尺度的改变 ,将非线性系统周期解的周期值显式地出现在非线性系统的系统方程中 ,然后对传统打靶法进行改造 ,把周期值T作为一参数 ,通过优化方法选择出迭代过程中增量的改变值 ,T也参入打靶过程。求解过程既包含对周期轨道的求解 ,又包含对周期T的求解 ,从而能迅速准确地求出周期轨道和周期T。文中用该方法对VanDerPol方程和非线性转子—轴承系统进行了求解 ,验证了方法的有效性。

A generalized shooting method for determining periodic orbit and its period of nonlinear dynamical system is suggested. By changing the time scale τ=t/T , in which T and t denote the period and time of original system separately, τ denotes time of the system transformed, the period of orbit of the original nonlinear system is drawn into the governing equation of the new system. The period of system transformed is 1. The value of period is used as one of the parameters to take part in the iteration. Shooting method modified in this paper assumes a point x 0 in the periodic solution and period T 0, and then shoots in time from 0 to 1. Then x 1 which is the value in τ =1 can be obtained. Order r=x 1-x 0 , and checks weather r is satisfied with precession requested. If r satisfies with precession requested then stop, otherwise, using Newton Fox method to work out increment of x 0 and T 0. Get the partial derivative of x and T about r , thus a linear equation sets of n equations of n +1 variable (Δ x , Δ T ) will be formed. As the number of variable is more than the number of equations, there are innumerable group solutions. In order to solve the values of Δ x and Δ T from the equations mentioned above, one variable must be fixed. In terms of the difference between the n +1 initial conditions selected and the value of period at end point obtained after one period integral, the variable that must be fixed can be selected. If r i is the least one in r , that means the initial value selected is closest to the actual periodic orbit of the system. Therefore, we choose the minimum r i worked out forward to be fixed. But if r i which is related to T is the minimum, it can not be fixed because the period T of the periodic solution of the system is certain. Then the initial condition related to it will be keep constant at next iterative process. After working out the value Δ x and Δ T , order x=x +Δ x and T 0=T 0+Δ T , repeat such procedure until the precision requested is obtained. Thus the periodic orbit and period value of nonlinear system can be determined simultaneously. Lastly, the validity of such method is verified by determining the periodic orbit and period value for Van Der Pol equation and nonlinear rotor bear system.