Hierarchical structure in sharply divided phase space for the piecewise linear map

We have studied a two-dimensional piecewise linear map to examine how the hierarchical structure of stable regions affects the slow dynamics in Hamiltonian systems. In the phase space there are infinitely many stable regions, each of which is polygonal-shaped, and the rest is occupied by chaotic orbits. By using symbolic representation of stable regions, a procedure to compute the edges of the polygons is presented. The stable regions are hierarchically distributed in phase space and the edges of the stable regions show the marginal instability. The cumulative distribution of the recurrence time obeys a power law as $\sim t^{-2}$, the same as the one for the system with phase space, which is composed of a single stable region and chaotic components. By studying the symbol sequence of recurrence trajectories, we show that the hierarchical structure of stable regions has no significant effect on the power-law exponent and that only the marginal instability on the boundary of stable regions is responsible for determining the exponent. We also discuss the relevance of the hierarchical structure to those in more generic chaotic systems.

Figure 1

(a) Phase space portrait for the piecewise linear map with $K=1$. (b) The two invariant domains are denoted by ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$. The red lines indicate the boundary between these two regions. (c) Magnification of the phase-space portrait. (d) Phase space partitioning for the map (2) with $K=1$. Each region is defined by the inequalities (4).

Figure 2

(a) Subregions for symbol length $n=2$. The boundaries of the subregions are calculated by the procedure presented in the Appendix. (b) Subregions for $n=3$. (c) The type of subregions for $n=2$. There are only hyperbolic and elliptic regions. (d) The type of subregions for $n=3$.

Figure 3

(a) Stable regions with symbol length 3, 6, 15, 33, 51, 69, and 87. The number of the regions is 1, 2, 10, 66, 204, 552, and 1218, respectively. Except for symbol length 3 cases, every stable region has its symmetric one with respect to the line $x=y$. (b–d) Magnification of the stable regions. The stable regions are hierarchically aligned.

Figure 4

Cumulative recurrence-time distributions. The inset shows the phase space and the recurrence regions. The recurrence region $T_1$ is a rectangular ($0.1\le x\le 0.2,0.1\le y\le 0.2$) and $T_2$ is ($0.3\le x\le 0.48,0.01\le y\le 0.02$). $T_1$ is contained in ${\mathcal{I}}_1$ and $T_2$ in ${\mathcal{I}}_2$. The recurrence regions are chosen so as not to overlap any stable regions.

Figure 5

A typical recurrence orbit for (a) the recurrence region $T_1$ and for (b) $T_2$. Stable regions are shown as filled black polygons. The recurrence time of the orbits is 1148 for (a) and 1044 for (b).

Figure 6

(a) Symbol sequence statistics $S\left(\tau \right)$ for the recurrence region $T_1$. The length of the symbol sequences is 3. 26 symbol sequences are depicted. (b) Semilogarithmic plot of (a). (c) Symbol sequence statistics for the symbol sequence of length 6. The symbol sequences for the power law are classified into two groups, ${\mathcal{F}}_{ead}=\{eadead,adeade,deadea,daedae,edaeda,eadead\}$ and ${\mathcal{F}}_{afabdb}=\{afabdb,bafabd,dbafab,bdbafa,abdbaf,fabdba\}$. (d) Symbol statistics for ${\mathcal{F}}_{ead}$ (${\mathcal{F}}_{dae}$) and ${\mathcal{F}}_{afabdb}$ with several symbol lengths. As the symbol length increases, the symbol statistics asymptotically approaches a power-law tail. The prefactor for the power law tail for ${\mathcal{F}}_{ead}$ is slightly greater than the one for ${\mathcal{F}}_{afabdb}$.

Figure 7

(a) Phase-space regions in which the power-law behavior is found in the symbol sequence statistics. The black filled hexagon represents the stable region and the boundary lines between ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ are indicated by the black lines. (b) Magnification of panel (a). The regions with different symbol length are overlaid. As the symbol length $l$ increases, the region of ${\mathcal{F}}_{ead}$ shrinks into the boundary lines. For ${\mathcal{F}}_{afabdb}$ the region shrinks into the hexagon edges as the symbol length increases.

Figure 8

Symbol statistics for the recurrence region $T_2$. The symbol length is 39. The number of initial points is taken as ${10}^8$.

Figure 9

Phase space regions in which the power-law behavior is found in the symbol sequence statistics. (a) The whole phase space. From (b) and (c) to (d), we gradually magnify hierarchical regions. The black filled polygons show stable regions.

Figure 10

Schematics picture of the hierarchical structures in (a) generic mixed phase space and in (b) our sharply divided mixed phase space. The black regions indicate the stable regions and the gray ones indicate the surrounding sticky regions.