Bouncing ball on a vibrating periodic surface

We present an investigation of a partially elastic ball bouncing on a vertically vibrated sinusoidal surface. Following the work of McBennett and Harris [Chaos 26, 093105 (2016)], we begin by demonstrating that simple periodic vertical bouncing at a local minimum of the surface becomes unstable when the local curvature exceeds a critical value. The resulting instability gives rise to a period doubling cascade and results in persistent horizontal motion of the ball. Following this transition to horizontal motion, periodic "walking" states-where the ball bounces one wavelength over each vibration cycle-are possible and manifest for a range of parameters. Furthermore, we show that net horizontal motion in a preferred direction can be induced by breaking the left-right symmetry of the periodic topography.