Improvement of the calculation method for the grazing bifurcation point
in two-dimensional impact oscillator <Abstract> Impact oscillator have attracted attention in various fields since old times. Also it is known that the grazing bifurcation can be observed in the impact oscillator. Grazing bifurcation occurs when the trajectory tangentially hits the border and affects the system dynamics. Thus, it is important to clarify the mechanism of such phenomenon. In recent years, researchers have proposed some calculating methods of the grazing bifurcation point. However, the composition of the Jacobian matrix is complicated, because these methods treat the periodic time of the solution as a dependent variable. This paper proposes a new method for calculating the grazing bifurcation point in two-dimensional impact oscillator. Our method has the property that the periodic time is treated as an independent variable. First, we explain the two-dimensional impact oscillator with fixed border. When the trajectory reaches one boundary, the solution immediately jumps to the other. Next, we express the calculation method of the grazing bifurcation point using some conditional equations. Finally, we confirm the validity of our method using the two parameter bifurcation diagram, which is obtained by applying the method to Rayleigh-type oscillator. |