Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction

Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packets, coherent states, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentially with excursion complexity, and the corresponding cell areas decrease exponentially in size as well. With the specification of a desired precision, the exponentially proliferating set of homoclinic orbit actions is expressible by a slower-than-exponentially increasing set of cell areas, which may present a means for developing greatly simplified semiclassical formulas.

Figure 1

Horseshoe-shaped homoclinic tangle formed by $S\left(x\right)$ (red dashed curve) and $U\left(x\right)$ (black solid curve), with two primary homoclinic orbits $\left\{h_0\right\}$ and $\left\{g_0\right\}$. Notice that the $U\left(x\right)$ segments beyond $g_0$ and $h_1$ are simply connected and omitted from the figure for clarity, and the same for the $S\left(x\right)$ segments beyond $g_{-1}$ and $h_{-1}$.

Figure 2

Upper panel: Trellis $T_{-1,2}$. Lower panel (zoomed-in): The hierarchy of homoclinic points on $S_{-1}^{\prime }$ can be organized as the following: the winding-1 points $g_{-2}$ and $h_{-1}$ constitute the first-order family. The sequence of winding-2 points $a^{\left(n\right)}$ and $b^{\left(n\right)}$ ($n\ge 0$) form two second-order families that accumulate on $g_{-2}$ asymptotically under Eqs. (9) and (10). Similarly on the right side of $S_{-1}^{\prime }$, we have the winding-2 points $c^{\left(n\right)}$ and $d^{\left(n\right)}$ ($n\ge 1$) accumulating on $h_{-1}$, which form two second-order families as well. Consequently, three families of areas $\left[{\mathcal{A}}_{SUSU[g_{-2},b^{\left(n\right)},d^{\left(n\right)},h_{-1}]}^{\circ }\right], \left[{\mathcal{A}}_{SUSU[b^{\left(n\right)},a^{\left(n\right)},c^{\left(n\right)},d^{\left(n\right)}]}^{\circ }\right]$, and $\left[{\mathcal{A}}_{SUSU[a^{\left(n\right)},b^{(n-1)},d^{(n-1)},c^{\left(n\right)}]}^{\circ }\right]$ accumulate on the bottom segment $U[g_{-2},h_{-1}]$ under Eq. (16), with the same asymptotic exponent ${\mu }_0$.

Figure 3

Accumulation of homoclinic points along $U\left(x\right)$ under $M^{-1}$. Two families of homoclinic points $\left[v^{(-n)}\right]$ and $\left[w^{(-n)}\right]$ are created from $S_{-n}^{\prime }\cap U_0$, that accumulate on $g_0$ along $U_0$. Notice that only the $n=1,2$ cases are plotted here. Similarly, the two families $\left[y^{(-n)}\right]$ and $\left[z^{(-n)}\right]$ are created from $S_{-n}^{\prime }\cap U_1$, and accumulate on $h_1$ along $U_1$.

Figure 4

Partitioning of cell areas in $T_{-1,1}$. The three type-I cells are $A_{\alpha }, A_{\beta }$, and $A_{\gamma }$. The three type-II cells are $B_{\alpha }, B_{\beta }$, and $B_{\gamma }$. The $A$ cell from $T_{-1,0}$ is partitioned into three cell areas in $T_{-1,1}$: $A=A_{\alpha }+A_{\beta }+A_{\gamma }$. Similarly for the type-II cell, $B=B_{\alpha }+B_{\beta }+B_{\gamma }$.

Figure 5

Zoomed-in graph around the complex region of $T_{-1,2}$ (same as the lower panel of Fig. 2). The $A_{\alpha }$ and $A_{\gamma }$ areas in Fig. 4 are partitioned into three subareas: $A_{\alpha }=A_{\alpha \alpha }+A_{\alpha \beta }+A_{\alpha \gamma }$ and $A_{\gamma }=A_{\gamma \alpha }+A_{\gamma \beta }+A_{\gamma \gamma }$. The $A_{\beta }$ area does not get partitioned because of the open system assumption, i.e., manifolds outside of the complex region do not revisit the complex region in future iterations. Since the type-I and type-II cells are always partitioned by any lobe $L_n$ simultaneously, the $B_{\alpha }$ and $B_{\gamma }$ cells from Fig. 4 are partitioned in identical ways with $A_{\alpha }$ and $A_{\gamma }$, respectively.

Figure 6

(Schematic) The cell $A_{\tilde{\omega }}$ in $T_{-1,n}$ is partitioned into three new cells in $T_{-1,n+1}$ by lobe $L_{n+1}$: $A_{\tilde{\omega }}=A_{\tilde{\omega }\alpha }+A_{\tilde{\omega }\beta }+A_{\tilde{\omega }\gamma }$, where $A_{\tilde{\omega }\beta }\subset L_{n+1}$. The rule of assignment is, $A_{\tilde{\omega }\beta }$ is always assigned to the middle cell, and $A_{\tilde{\omega }\alpha }$ is assigned to the cell with the two corners, namely $a_{\alpha }$ and $b_{\alpha }$, such that ${\displaystyle a_{\beta },c_{\beta }\underset{S}{\overset{k}{\hookrightarrow }}{a}_{\alpha }}$ and ${\displaystyle b_{\beta },d_{\beta }\underset{S}{\overset{k}{\hookrightarrow }}{b}_{\alpha }}$, where the order number $k$ is an appropriate integer that depends on $\tilde{\omega }$. Finally, $A_{\tilde{\omega }\gamma }$ is assigned to the last cell. The same pattern applies to all the $B$ cells as well.

Figure 7

(Zoomed-in graph) The $A_{\gamma }$ and $B_{\gamma }$ cells from Fig. 4 are partitioned by $L_2$ into three new cells each. The $A_{\gamma \beta }$ area is assigned to the middle one. Since ${\displaystyle v,w\underset{S}{\overset{1}{\hookrightarrow }}{g}_0}$ and ${\displaystyle r^{\left(1\right)},s^{\left(1\right)}\underset{S}{\overset{1}{\hookrightarrow }}{b}^{\left(0\right)}}, A_{\gamma \alpha }$ is assigned to the top one, leaving $A_{\gamma \gamma }$ to be the bottom one. The same rules apply to the $B$ cells as well.

Figure 8

The partition tree of type-I cell areas. Nodes at the $n\mathrm{th}$ level along the tree are areas generated from the partition of $T_{-1,n-1}$ by $T_{-1,n}$. Every $\alpha$ and $\gamma$ nodes are partitioned into three new nodes at the next level, while the $\beta$ nodes do not get partitioned any further. The partition tree of type-II areas follows an identical pattern upon changing the symbols $A$ into $B$. To order the cells as in the trellis, proceeding from the top of the tree, reverse the order for the next level down each time the number of $\gamma$ symbols is odd.

Figure 9

(Schematic) Successive partitions of $A_{\tilde{\omega }}\subset T_{-1,n}$ in later trellis $T_{-1,n+m}$. Upper panel ($T_{-1,n+1}$): The same as Fig. 6, where $A_{\tilde{\omega }}$ is partitioned into three areas by $L_{n+1}$. Middle panel ($T_{-1,n+2}$): Zoomed-in graph of $A_{\tilde{\omega }\alpha }$ in $T_{-1,n+2}$, where $A_{\tilde{\omega }\alpha }$ is partitioned by $L_{n+2}$ into three new areas. Lower panel ($T_{-1,n+3}$): Zoomed-in graph of $A_{\tilde{\omega }\alpha \alpha }$ in $T_{-1,n+3}$, where $A_{\tilde{\omega }\alpha \alpha }$ is partitioned by $L_{n+3}$ into three new areas. The addition of successive lobes create four families of homoclinic points, $\left[a_{{\alpha }^{m-1}\beta }\right], \left[c_{{\alpha }^{m-1}\beta }\right], \left[b_{{\alpha }^{m-1}\beta }\right]$, and $\left[d_{{\alpha }^{m-1}\beta }\right]$, that accumulate on $a_{\alpha }$ and $b_{\alpha }$ under Eqs. (24) and (25) with exponent ${\mu }_0$. Therefore, the three families of areas $\left[A_{\tilde{\omega }{\alpha }^m}\right], \left[A_{\tilde{\omega }{\alpha }^{m-1}\beta }\right]$, and $\left[A_{\tilde{\omega }{\alpha }^{m-1}\gamma }\right]$ also converge onto $U[a_{\alpha },b_{\alpha }]$ with exponent ${\mu }_0$, as described by Eq. (23).

Figure 10

An example of the homoclinic orbit action decomposition. As shown by Eq. (37), the relative action of the winding-2 orbit $\left\{d^{\left(1\right)}\right\}$ is decomposed into the sum of the relative actions of the winding-1 orbits $\left\{h_{-1}\right\}$ and $\left\{g_1\right\}$, and a phase-space area ${\mathcal{A}}^{\circ }\left(d^{\left(1\right)}\right)={\mathcal{A}}_{SUSU[d^{\left(1\right)},h_{-1},x,g_1]}^{\circ }$ marked by the hatched region in the figure. Similar decomposition can be done for any homoclinic point on $S_{-1}^{\prime }$.

Figure 11

Relative areas for the decomposition of the winding-3 orbit $\left\{r^{\left(1\right)}\right\}$. The ${\mathcal{A}}^{\circ }\left(r^{\left(1\right)}\right)$ term in Eq. (43) is marked as the hatched region in the lower panel, which is just $-A_{\gamma \alpha }$. The long and curvy, hatched region in the upper panel is the ${\mathcal{A}}^{\circ }\left(v\right)$ term in Eq. (45). Areas like this may not be expressible by the type-I and type-II cell areas.

Figure 12

(Schematic) Homoclinic tangle forming an incomplete horseshoe. Comparing to the complete horseshoe case (Fig. 2), the points $a^{\left(0\right)}, b^{\left(0\right)}, c^{\left(1\right)}$, and $d^{\left(1\right)}$ are pruned. However, the accumulation relations (thus the projection operations) for the unpruned homoclinic points remain the same. Therefore, for the unpruned homoclinic points, Eqs. (35) and (49) remain valid.

Figure 13

The logarithmic dependence of $d$ with respect to $N$ using the slow scaling exponent ${\mu }_1=1.483$, error tolerance $\delta \mathcal{F}/A=0.001$, and $a=10$ for the Hénon map. For the computation of large trellises such as $N=100$, the approximate formula of Eq. (55) only requires the computation of cell areas up to trellis number $d=8$.

Figure 14

Trellis $T_{-1,1}$. The fundamental segments $U_0$ and $S_0^{\prime }$ are indicated by thick solid and thick dashed curve segments, respectively. The lobes $L_0$ and $L_0^{\prime }$ (hatched regions) form the turnstile which governs the phase-space transport.

Figure 15

Smale horseshoe formed by $S\left(x\right)$ (red dashed curve) and $U\left(x\right)$ (black solid curve). The vertical strips $V_0$ and $V_1$ in the upper panel (hatched regions) are the generating partitions of the symbolic dynamics, and are mapped into the horizontal strips $H_0$ and $H_1$ in the lower panel (hatched regions), respectively, under one iteration. The fixed point $x$ has symbolic string $\overline{0}.\overline{0}$, and the primary homoclinic points $h_0$ and $g_0$ have symbolic strings $\overline{0}1.1\overline{0}$ and $\overline{0}1.\overline{0}$, respectively.

Figure 16

Homoclinic points in $T_{-1,0}$. The symbolic codes are $h_{-1}\Rightarrow \overline{0}.11\overline{0}, g_{-2}\Rightarrow \overline{0}.01\overline{0}, a^{\left(0\right)}\Rightarrow \overline{0}1.11\overline{0}$, and $b^{\left(0\right)}\Rightarrow \overline{0}1.01\overline{0}$. The hierarchical relations are ${\displaystyle a^{\left(0\right)},b^{\left(0\right)}\underset{S}{\overset{1}{\hookrightarrow }}{g}_{-2}}$. Notice that the hierarchical relations are indicative for the assignments of symbolic codes: the codes of $a^{\left(0\right)}$ and $b^{\left(0\right)}$ can be obtained by adding the substrings “11” and “10”, respectively, to the left end of the core of $g_{-2}$, while maintaining the position of the decimal point relative to the right end of the core.

Figure 17

Homoclinic points in $T_{-1,1}$. The symbolic codes are $a^{\left(1\right)}\Rightarrow \overline{0}11.01\overline{0}, b^{\left(1\right)}\Rightarrow \overline{0}10.01\overline{0}, c^{\left(1\right)}\Rightarrow \overline{0}11.11\overline{0}$, and $d^{\left(1\right)}\Rightarrow \overline{0}10.11\overline{0}$. The hierarchical relations are ${\displaystyle a^{\left(1\right)},b^{\left(1\right)}\underset{S}{\overset{2}{\hookrightarrow }}{g}_{-2}}$ and ${\displaystyle c^{\left(1\right)},d^{\left(1\right)}\underset{S}{\overset{1}{\hookrightarrow }}{h}_{-1}}$.

Figure 18

(Schematic) Iterates of a curve intersecting the stable manifold approach the unstable manifold. Future iterations of the curve ${\mathcal{C}}_0$ create a family of curves $\left[{\mathcal{C}}_n\right]$, which intersect $\overline{\mathcal{C}}$ at a family of points $\left[z^{\left(n\right)}\right]$. $\left[z^{\left(n\right)}\right]$ accumulates on $z_u$ under the exponent ${\mu }_x$, as given by Eq. (D1).