2012/03/19 by Andrzej Okniński, Andrzej Okninski, B. Radziszewski +1
Engineering · Mathematics · Physics and Astronomy · #Ball (mathematics) #Bifurcation #Chaotic #Classical mechanics #Computation #Computer science #Experimental and Theoretical Physics Studies #Fixed point #Limiter #Mathematical analysis #Mathematics #Mechanics #Nonlinear system #Period-doubling bifurcation #Periodic function #Physics #Poincaré map #Quantum chaos and dynamical systems #Sports Dynamics and Biomechanics #msc:70B05 #msc:70F35 #nlin.CD
paper · pdf · doi:10.1007/s12591-012-0137-3
published as Differ. Equ. Dyn. Syst. 21, 165-171 (2013) · 8 pages, 1 figure, presented at the DSTA 2011 conference, Lodz, Poland
arxiv created 2012/03/19 · openalex publication_date 2012/07/31 · arxiv updated 2013/02/12 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/29
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare 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, 2 - 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.