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

Abstract

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2-cycles, grazing and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2-cycles in a corner-type bifurcation.

Simple Model of Bouncing Ball Dynamics

“…Accordingly, we have decided to choose the limiter's periodic motion in a polynomial form to make analytical investigations of the dynamics possible. In our previous papers we have assumed displacement of the table as piecewise linear periodic function of time [9,10] as well as quadratic [11]. In this work we study dynamics for a cubic function of time Y c (T ): In Fig.…”

“…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].…”

Non-stationary dynamics in the bouncing ball: A wavelet perspective

“…16 Okninski and Radziszewski, have studied numerically, as well as analytically, the dynamics of a bouncing ball moving in a gravitational field and colliding with a vertically moving table with constant velocity. 17 This driven system shows non-stationary behavior and unstable periodic orbits. It also exhibits chaotic dynamics under appropriate parametric conditions.…”

Bouncing ball dynamics: Simple model of motion of the table and sinusoidal motion

“…It is thus possible to approximate the sinusoidal motion of the table more and more exactly and conduct analytical computations. Carrying out this plan we have studied several such models with linear, quadratic and cubic polynomials [16,17,18,19].…”