Existence Of The Horseshoe Of The Bouncing Ball Model

The Bouncing Ball model which was raised firstly by Meriam in1975 possesses rich dynamic behavior, many scholars had studied itsuch as Wood and Byrne in 1981[3], and Holmes in 1982[4]. In addition , Guckenheimer and Holmes spent a great of length to discuss itin the book [1] in 1983. They gave the sufficient condition γ ≥ 5π that Ensure the existence of the horseshoe of the model for α= 1 and used numerical value method to discovered "stranfe attractor " for certain a (for example : α - 1, γ - 6π) where α γ denote the colliding coefficient of restitution and the amplitude of the Ball. Recently , XieJianhua~[14] obtained there is a hyperbolic invariant set when γ≥γmm = 4.0318π on the basis of the symmetry of the model for α = 1 .It is clear that the scope of the coefficient α is 0<π≤1 form book [ 1 ] . So it is necessary to research the existence of horseshoe of the model for 0 < α < 1.In this paper , we choose a reasonable basic area and a coordinate transform and give conditions that ensure the invariant set or hyperbolic invariant set exist in the Bouncing Ball model by using the Mosers and Zhou Jianying's conditions respectively . Thus, the existence of horseshoe of the Bouncing Ball model is solved effectively...