An analytical and numerical study of chaotic dynamics in a simple bouncing ball model

Abstract

Dynamics of a ball moving in gravitational field and colliding with a moving table is studied in this paper. The motion of the limiter is assumed as periodic with piecewise constant velocity-it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity. The Poincaré map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically and numerically.

Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

“…This transition is a corner-type bifurcation similar to that found in a bouncing ball model with piecewise linear velocity [15]. In our future work, we shall study models with displacement of the table described by cubic functions of time [16].…”

“…Secondly, equations for a stable 2-cycle were found and simplified, cf. (14), (15). From these equations, the analytical condition for birth of the 2-cycle was found (cf.…”

“…In our previous papers, we have assumed displacement of the table as piecewise linear periodic function of time [12][13][14][15]. In our recent work, preliminary results for function Y (T ) assumed as quadratic, Y q , and two cubic functions of time, Y c 1 and Y c 2 , have been obtained [16].…”

“…Recently, we have considered several models of motion of a material point in a gravitational field colliding with a limiter moving with piecewise constant velocity [12][13][14][15]. Moreover, we have proposed more realistic yet still simple models approximating sinusoidal motion of the table as exactly as possible but still preserving possibility of analytical computations [16].…”

Simple Model of Bouncing Ball Dynamics

“…We approached this problem assuming a special motion of the table. Recently, we have considered several models of motion of a material point in a gravitational field colliding with a limiter moving periodically with piecewise constant velocity [9,10] and velocity depending linearly on time [11]. In the present work we study the model in which periodic displacement of the table is a cubic function of time, carrying out our project to approximate the sinusoidal motion of the table as exactly as possible but preserving possibility of analytical computations [12].…”