A BREATHING CHAOS

The dynamics of a billiard in a gravitational field between a vertical wall and an inclined plane depends strongly on the angle theta between wall and plane. Most conspicuously, the relative amount of chaotic versus regular parts of the energy surface shows pronounced oscillations as a function of theta , with distinct minima for theta near 90 degrees /n (n=2, 3,. . .). This breathing is also seen in the Lyapunov exponents. It reflects a repetitive pattern in the linear stability properties of families of periodic orbits. To study these orbits and their stability, Birkhoff's decomposition of the Poincare map into the product of two involutions is employed. The breathing in the amount of chaos can then be discussed in terms of the topology of symmetry lines, and of the corresponding directions of reflection.