Accurate shooting methods for computing periodic solutions of nonlinear systems

Abstract

AbstractThe shooting method is an effective technique for computing periodic solutions of nonlinear systems. In this method, a system of nonlinear integral equations is formulated which has the periodic solution as the solution. Then an iterative method such as Newton's method is applied to the system. However, when high accuracy is required in the numerical solution, a large amount of computation time is necessary because fine steps generally are used for the numerical integration. This paper discusses the application of the Richardson extrapolation to the problem of computing accurate periodic solutions of nonlinear systems. First, it is shown that the error of the numerical solution computed by the shooting method has an asymptotic expansion if the trapezoidal rule is used in the numerical integration. This is easily proved by introducing Keller's theory which is concerned with the finitedifference method. Then it is shown that it is possible to improve the accuracy of the periodic solution significantly by applying the Richardson extrapolation to several numerical solutions with different step sizes. It is also proved that Newton's method converges quadratically if the periodic solution is isolated and the step sizes are sufficiently small. Some numerical examples are given to show the effectiveness of the approach presented here.