A new family of periodic breather wave solutions and kink breather wave solutions of initial perturbation for the (1+1)-dimensional Benjamin-Ono(BO) equation are obtained by using the homoclinic test method with bilinear transformation of initial perturbation. Meanwhile, rational wave solutions of initial perturbation and bifurcation point of dynamics are also gained by applying quadratic function ansatz method. Some local perturbation structure of dynamics are shown directly. Results show that the dynamical behavior of the equation is sensitive to initial values.