Chaotic Motion of a Bouncing Ball on a Harmonically Forced Oscillator

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This Demonstration shows the map of the movement of a ball bouncing on a harmonically forced oscillator, depending on the frequency, the forcing amplitude, and the coefficient of restitution. The coordinates of each point are given by the dimensionless velocity of the ball after the collision and by the phase of the oscillator.

Contributed by: Sotér Gábor (November 2012)
Open content licensed under CC BY-NC-SA



The equations of the map are




ϕ is the phase of the oscillator,

is the dimensionless velocity of the ball after the collision,

is the driving frequency,

is the forcing amplitude,

is the gravitational acceleration,

is the coefficient of restitution.


[1] P. J. Holmes, "The Dynamics of Repeated Impacts with a Sinusoidally Vibrating Table," Journal of Sound and Vibration, 84(2), 1982 pp. 173–189.

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