Differential drive robots, such as robotic vacuums, often have at least two motion primitives: the ability to travel forward in straight lines, and rotate in place upon encountering a boundary. They are often equipped with simple sensors such as contact sensors or range finders, which allow them to measure and control their heading angle with respect to environment boundaries. We aim to find minimal control schemes for creating stable, periodic 'patrolling' dynamics for robots that drive in straight lines and 'bounce' off boundaries at controllable angles. As a first step toward analyzing high-level mobile robot dynamics in more general environments, we analyze the location and stability of periodic orbits in regular polygons. The contributions of this paper are: 1) proving the existence of periodic trajectories in n-sided regular polygons and showing the range of bounce angles that will produce such trajectories; 2) an analysis of their stability and robustness to modeling errors; and 3) a closed form solution for the points where the robot collides with the environment boundary while patrolling. We present simulations confirming our theoretical results.