Prevalence of marginally unstable periodic orbits in chaotic billiards

The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and stands apart from the regular regions. We show that these structures both exist and strongly influence the dynamics of locally perturbed billiards, which include a large class of widely studied systems. We demonstrate the impact of these structures in the quantum regime using microwave experiments in annular billiards.

Figure 1

Adding local perturbations to integrable billiards, as those shown in (a)–(c), one obtains frequently studied chaotic billiards, such as those shown in (d)–(h). The gray regions of the (a) rectangular, (b) circular, and (c) elliptical billiards are defined in such a way that chaotic motion is possible [$a<R$ in (b) is the radius of the smallest circle that circumscribes all scatterers]. MUPOs are shown here to exist in billiards such as (d) Sinai [1], (e) stadium [2], (f) annular [10], (g) mushroom [11], and (h) elliptical with scatterers [12].

Figure 2

(Color online) Annular billiard for parameters $r=0.35$ and $\delta =0.5$: (a) configuration space and (b) phase space. MUPOs correspond to the periodic orbits that cross the circle of radius $a$ [dotted line in (a)] but that do not hit the scatterer [region between the dashed lines in (b) in which relation (1) is satisfied]. The symbols $◼$ and $▴$ indicate, respectively, individual orbits belonging to MUPOs $(2,1)$ and $(5,1)$.

Figure 3

(Color online) Size $w$ of the families of MUPOs $(5,2)$ in the annular billiard with $r=0.12$: (a) outer MUPOs for $\delta =0.2$, (b) mixed inner-outer MUPO for $\delta =0.5$, and (c) inner MUPOs for $\delta =0.8$. All three kinds of MUPOs may coexist for a fixed $\delta$. The size $w$ of the families of MUPOs in Eq. (3) is given by the length of the external arcs. MUPOs $(5,2)$ outside of these regions do not exist since they collide with the inner scatterer.

Figure 4

Experimental length spectrum of the annular billiard for $\delta =0.48$. Peaks associated with four periodic orbits are indicated: the shortest one is unstable, the diameter and triangle are MUPOs, and the square is inside the regular region.

Figure 5

(Color online) Semiclassical strengths $S_{\mathrm{sc}}$ (lines) compared to the experimental strengths $y$ (symbols). $S_{\mathrm{sc}}$ is given by Eq. (7) and expected from periodic orbit theory, while $y$ is extracted from the length spectra, as shown in Fig. 4. Three orbits are considered: diameter (dotted line and circles), triangle (solid line and triangles), and square (dot-dashed line and squares). The horizontal lines (top) indicate the values of $\delta$ for which the corresponding MUPOs exist.