Development of shooting method for impact systems

Abstract

A shooting method is a very powerful numerical method to obtain periodic solutions of nonlinear systems. However, as a variational equation of motion is needed in the shooting method and it is very difficult to obtain it in the impact systems, the shooting method for impact systems has not been developed. In this report, a shooting method for impact systems is presented by solving this problem of variational equation. Namely, the variational equation with the delta function and its differentiation is derived. It is shown that the calculation speed of this method is very fast and complicated periodic solutions are easily obtainable in high accuracy. The stabilities of periodic solutions obtained in the shooting method are in good accordance with those obtained by the analytical method. The discontinuities in the stability of the periodic solutions are shown using characteristic multiplier. Lyapunov exponents are also calculated by applying the integral technique of variational equation.