A numerical approach to calculate grazing bifurcation points in an impact oscillator with periodic boundaries

In this paper, we propose a numerical method to calculate the grazing bifurcation points in an impact oscillator with periodic boundaries. First, we illustrate the n-dimensional autonomous impact oscillator with the moving boundaries. The boundaries consist of the scalar function involving the periodic function. When the trajectory hits one boundary, the solution jumps to the other immediately. Next, we construct the composite Poincaré map and show its derivatives. Using the derivatives, we calculate the location of the periodic point, bifurcation parameter and time that elapses before the trajectory hitting the boundary. Finally, we apply the method to a Rayleigh-type oscillator with the sinusoidal boundaries to confirm its validity.