ABSTRACT
Viscous Case For this case, we assume that collisions are completely elastic ( 1=α ), and there is no tangential friction between the particle and the surface ( 0=β ), but the media above the surface is viscous fi xing the value of h=0.1 as in [22]. We use the reduced impact representation, where the height of the bouncing ball is sampled at each impact with the surface (impact sampling). Since the system is dissipative, we plot the bouncing process after the initial transients cease down _see Fig. 5(a). We skip 1500 successive bounces before starting to plot the collision heights ()tv ˆ for every discrete value of ω . Parameter ω is varied following the rule:
A phenomenological model could be used to exemplify the bifurcation diagram presented in Fig. 5(a). The logistic map [24] is probably the simplest model ever used to study the transition to chaos via a period doubling route. Simple computational experiments with appropriately chosen parameter values of the logistic map would illustrate the universality of the bifurcation diagram in Fig. 5(a).
