Internal-wave billiards in trapezoids and similar tables

Abstract

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) minimality (all trajectories are dense); (2) there exist a global attractor and a global repellor, which are periodic and might coincide; (3) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories. Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) is a positive-measure fractal; the set for (2) has positive measure (giving a rigorous proof of the existence of Arnol'd tongues for internal-wave billiards); the set for (3) has measure zero.

On (n,1) Wave Attractors: Coordinates and Saturation Time

“…In Section 3 we show that it can be easily computed analytically and extended for integer n > 1. On the other hand, the shape of the area of existence of a more general (n,m) attractor with n horizontal and m > 1 vertical cells can have a more complex shape [25]. Note that attractors require m to be odd, as for even m a focusing reflection at inclined walls is exactly balanced by a defocusing reflection.…”

“…For example in [25] the shape of (1,3) and (2,3) regular modes is given. In the next section we will give exact expressions for the coordinates of boundary reflections of (n,1) attractors and areas of their existence on a (d, τ) diagram.…”

“…Figure 5 shows the application of this idea. See also a discussion on unfolding in [25,31] which is used to find the coordinates of the (1,1) attractor. First, we reflect the trapezium with respect to the left vertical boundary.…”

2D internal waves in an ergodic setting

“…The family of circle diffeomorphisms given by the chess billiard map b(•, λ) defined in (1.8) was also recently investigated by Lenci et al in trapezoidal domains [LBC21]. We mention that it is called the chess billiard map since it was originally used to study a generalization of the n-Queens problem by Hanusa-Mahankali [HM19], and has been further studied by Nogueira-Troubetzkoy [NT20].…”