Stability of periodic orbits in no-slip billiards

Abstract

Rigid bodies collision maps in dimension-two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards—planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change after collisions with the boundary of the billiard domain.This paper, which continues the investigation initiated in Cox and Feres (2017 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Chernov (Providence, RI: American Mathematical Society), is mainly focused on the issue of stability of periodic orbits in no-slip planar billiards. We prove Lyapunov stability of periodic orbits in polygonal billiards of this kind and, for general billiards domains, we obtain curvature thresholds for linear stability at commonly occurring period-2 orbits. More specifically, we prove that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than π, no-slip billiard maps admit elliptic period-2 orbits; (ii) polygonal no-slip billiards under this same corner angle condition always contain small invariant neighborhoods of the periodic point on which, up to smooth conjugacy, orbits of the return map lie on concentric circles; in particular the system cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of the Sinai billiard must contain linearly stable periodic orbits of period 2 and, more generally, we obtain a curvature threshold at which the period-2 orbits go from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in wedge and triangular billiards. Our linear stability results extend those of Wojtkowski for the no-slip Sinai billiard.