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

Abstract

Considering the numerical continuation of periodic solutions, an efficient algorithm is proposed. This algorithm is based on the Broyden’s 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’s method here includes the steps of linear search and the QR (quadrature rectangle) decomposition to solve the linear equations. For the general periodic solutions, the period as a parameter to be continued is included in the periodicity conditions. 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’s method iteratively until the initial values converge. According to the property that the orbit passing across 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 conditions are obtained, and then solved by the Broyden’s method. Associating with the symmetry of the Hamiltonian system and some classifications of the periodic orbits of the PCRTBP, 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.