Differential Equations and Dynamical Systems

, Volume 21, Issue 1–2, pp 165–171 | Cite as

Simple Model of Bouncing Ball Dynamics

Displacement of the Limiter Assumed as a Cubic Function of Time
  • Andrzej OknińskiEmail author
  • Bogusław Radziszewski
Original Research


Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincaré map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, two cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter’s motion making analysis of chattering possible.


Bouncing ball Simple models Exact solutions 


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Copyright information

© Foundation for Scientific Research and Technological Innovation 2012

Authors and Affiliations

  1. 1.Department of Management and Computer ModellingKielce University of TechnologyKielcePoland
  2. 2.Collegium Mazovia Innovative UniversitySiedlcePoland

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