Classical Dynamics of Particles and Systems · Problem 4-23

Gravitation — Marion & Thornton

Physics — Mechanics · chirikov-map-bouncing-ball

Problem

The motion of a bouncing ball, on successive bounces, when the floor oscillates sinusoidally can be described by the Chirikov map:

p_{n+1} = p_n - K\sin q_n

q_{n+1} = q_n + p_{n+1}

where

-\pi \leq p \leq \pi

and

-\pi \leq q \leq \pi

. Construct two-dimensional maps for

K = 0.8

, 3.2, and 6.4 by starting with random values of

p

and

q

and iterating them. Use periodic boundary conditions, which means that if the iterated values of

p

q

exceed

\pi

, a value of

2\pi

is subtracted and whenever they are less than

-\pi

, a value of

2\pi

is added. Examine the maps after thousands of iterations and discuss the differences.