Stickiness in Hamiltonian systems: From sharply divided to hierarchical phase space

We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one-parameter families of marginally unstable periodic orbits and is characterized by an exponent $\gamma =2$ for the asymptotic power-law decay of the distribution of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws, depending on the relative contribution of the primary and secondary structures of the hierarchy. Numerical verification of our main results are provided for area-preserving maps, mushroom billiards, and the newly defined magnetic mushroom billiards.

Figure 1

(Color online) (a) Phase-space portrait of the continuous sawtooth map for $K=\frac{3}{2}$. The dots correspond to ${10}^4$ iterations of a chaotic trajectory and the blank region corresponds to the regular island. The straight lines represent three different families of period-three MUPOs (see text), and the different symbols correspond to specific MUPOs in one of these families. (b) Phase-space portrait of maps (2, 3, 4) for $K=2$. We plot $-0.5⩽y⩽0.5$ in (a) and $-0.5⩽x,y⩽0.5$ in (b) for visualization convenience. (c) From bottom to top, the cumulative RTDs of the maps considered in (a) and (b) (multiplied by a factor of 10 for clarity). The upper curve is a straight line with slope $\gamma =2$. Inset: distance of a chaotic trajectory to the border of the island in (a) during an event with recurrence time $T=335$.

Figure 2

Phase-space portrait of the continuous sawtooth map for $K=\frac{3}{2}$ showing the initial conditions of the trajectories that remain inside the dashed triangle for at least $n^{\prime }=1$, 2, 4 and 1000 iterations of the map, respectively. The inner triangle corresponds to the regular island. We plot $0⩽x⩽1.5$ and $-1⩽x⩽1$ for visualization convenience.

Figure 3

(Color online) (a) Typical mushroom billiard, defined by the geometric parameters $(r,R,h)$. Two MUPOs are shown in dotted lines. (b) Phase-space representation of the semicircular hat of the mushroom billiard with $r∕R=0.5$. (c) From bottom to top, RTDs for $r∕R=0.6$, 0.75, and 0.5 (multiplied by a factor of 2 for clarity). The upper curve is a straight line with slope $\gamma =2$. Inset: distance of a chaotic trajectory to the border of the island during an event with recurrence time $T=300$.

Figure 4

(Color online) (a) Illustration of the dynamics of a perturbed MUPO $(x_0,{\theta }_0+\epsilon )$ in the phase space (see text). (b) The corresponding dynamics in the configuration space of a billiard with parallel walls $(q=2)$.

Figure 5

Magnification of the phase-space portrait of the magnetic mushroom billiard at the border between the chaotic and regular regions for $r∕R=0.5$ and various values of the magnetic field.

Figure 6

Cumulative RTDs for the magnetic mushroom billiard with $r∕R=0.5$ and different values of the magnetic field. From bottom to top the lines represent: a power law with $\gamma =2$, the numerical results for $L=\infty$ (shifted downward by two decades for clarity), $L=100$ (shifted downward by one decade), and $L=50$, and a power law with $\gamma =1$.

Figure 7

Analysis of the magnetic mushroom billiard with $L=50$: (a) phase-space magnification at the border of chaos; (b) fraction $g\left(\theta \right)$ of recurrences that have $\theta$ as their minimal angle; (c) the RTD of all trajectories (upper solid curve), and the RTDs of the trajectories in regions (1, 2, 3, 4, 5) of (b) (lower solid curves). The lower curves in (c) are divided by 10 for clarity.

Figure 8

(Color online) Analysis of the magnetic mushroom billiard with $L=10$. (a) Phase-space magnification with two typical sticking trajectories with recurrence time $T\approx 8\times {10}^4$: trajectory 1 sticks near the upper island and trajectory 2 fills the chaotic region. (b) Fraction $g\left(\theta \right)$ of recurrences that have $\theta$ as their minimal angle. (c) The RTD of all trajectories (upper solid curve) and the RTDs of the trajectories in regions (1, 2, 3, 4, 5) of (b) (see legend). The lower curves in (c) are divided by 10 for clarity.