A ball that falls in a gravitational field before colliding against a flat surface will rebound from the surface with a loss of energy that depends on the coefficient of restitution. If the ball is free, it will continue bouncing on the surface in a series of collisions; these arise because in each collision the ball is partly elastic and during the period between collisions the ball is attracted towards the surface by gravity. In Chapter 2 it was shown that an inelastic ball (0 < e* < 1) which is bouncing on a level surface in a gravitational field has both a period of time between collisions and a bounce height that asymptotically approach zero as the number of collisions increases. In other words, with increasing time this dissipative system asymptotically approaches a stable attractor – the equilibrium configuration where the ball is resting on the level surface.

Some other systems can experience energy input during each cycle of impact and flight; consequently these systems exhibit more complex behavior. For example, a pencil has a regular hexagonal cross-section with six vertices. If the pencil rolls down a plane, the mean translational speed of the axis asymptotically approaches a steady mean speed of rolling where the kinetic energy dissipated by the collision of a vertex against the plane equals the loss in gravitational potential energy as the pencil rolls from one flat side to the next.