基于Broyden方法的不对称与对称周期解延拓算法与应用

An Algorithm on the Numerical Continuation of Asymmetric and Symmetric Periodic Orbits Based on the Broyden Method and Its Application

考虑周期解的数值延拓问题并提出基于Broyden拟牛顿法来延拓周期解的一种有效算法,先后以布鲁塞尔振子、平面圆型限制性三体问题(Planar Circular Restricted Three-Body Problem,PCRTBP)的周期解为例进行了验证.这里的Broyden方法包含线性搜索、正交三角分解求线性方程组的步骤.对一般的周期解,周期性条件方程组中含有周期作为待延拓参数,可用周期来决定积分时长,将解代入周期性条件得到积分型的非线性方程组,利用Broyden方法迭代延拓直至初值收敛.根据两次垂直通过一个超平面的轨道是对称周期轨道的性质,可采用插值的方法求得再次抵达超平面的解分量,得到周期性条件方程组,再用Broyden方法求解.结合哈密顿系统的对称性和PCRTBP周期轨道的一些分类,对2=1、3=1的内共振周期解族进行了数值研究.最后,对算法和计算结果做了总结和讨论.

Considering the numerical continuation of periodic solutions,an efficient algorithm is raised.This algorithm is based on the Broyden quasi-Newton method,and is verified by some examples of the periodic solutions of the Brusellator and the planar circular restricted three-body problem(PCRTBP).The Broyden method here includes the steps of linear search and the QR decomposition to solve the linear equations.For the general periodic solutions,the period as a parameter to be continued is included in the periodicity condition equations.The period can be used to determine the integration time,then the solution is substituted into the periodicity conditions to get the integral nonlinear equations,which are solved by using the Broyden method iteratively until the initial values converge.According to the property that the orbit passing through a hyperplane twice perpendicularly is a symmetric periodic orbit,the interpolation method can be used to obtain the solution components that reach the hyperplane again,and the periodicity condition equations are obtained,and then solved by the Broyden method.Associating with the symmetry of the Hamiltonian system and some classifications of the periodic orbits of the plane circular restricted three-body problem,a numerical study of the 2/1,3/1 internal resonant periodic solution families is carried out.Finally,the algorithm and calculation results are summarized and discussed.