Exact relations between homoclinic and periodic orbit actions in chaotic systems

Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulas expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This enables an explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.

Figure 1

(Schematic) Example partial homoclinic tangle from the Hénon map, which forms a complete horseshoe structure. The unstable (stable) manifold of $x$ is the solid (dashed) curve. There are two primary homoclinic orbits $\left\{h_0\right\}$ and $\left\{g_0\right\}$. $\mathcal{R}$ is the closed region bounded by loop ${\mathcal{L}}_{USUS[x,g_{-1},h_0,g_0]}$. Under forward iteration, the vertical strips $V_0$ and $V_1$ (including the boundaries) from the upper panel are mapped into the horizontal strips $H_0$ and $H_1$ in the lower panel. At the same time, points in region $E_0$ are mapped outside $\mathcal{R}$ into region $E_1$, never to return and escape to infinity. There is a Cantor set of points in $V_0$ and $V_1$ that remain inside $\mathcal{R}$ for all iterations, which is the nonwandering set $\mathrm{\Omega }$ defined in Eq. (A2). The phase-space itineraries of points in $\mathrm{\Omega }$ in terms of $V_0$ and $V_1$ give rise to symbolic dynamics, as described by Eq. (A3).

Figure 2

(Schematic) The correction term in Eq. (24) is determined by this exponentially small near-parallelogram area. The lower left corner $y_0=y_{n_{\gamma }}\Rightarrow \overline{{\gamma }^{+}{\gamma }^{-}}.\overline{{\gamma }^{+}{\gamma }^{-}}$ is a fixed point under $M^{n_{\gamma }}$. The other three corners are homoclinic points $h_{-n^{+}}^{\left(\gamma \gamma \right)}\Rightarrow \overline{0}{\gamma }^{-}.{\gamma }^{+}{\gamma }^{-}{\gamma }^{+}\overline{0}$, $h_0^{\left(\gamma \right)}\Rightarrow \overline{0}{\gamma }^{-}.{\gamma }^{+}\overline{0}$, and $h_{n^{-}}^{\left(\gamma \gamma \right)}\Rightarrow \overline{0}{\gamma }^{-}{\gamma }^{+}{\gamma }^{-}.{\gamma }^{+}\overline{0}$ . The unstable and stable segments between corresponding homoclinic points are labeled by $U$ and $S$, respectively. $C$ is a straight line segment connecting $h_{-n^{+}}^{\left(\gamma \gamma \right)}$ and $y_0$, which is exponentially close to the stable direction of $y_0$. Under $M^{n_{\gamma }},C$ is mapped into a near-straight segment $C^{\prime }$ connecting $h_{n^{-}}^{\left(\gamma \gamma \right)}$ and $y_{n_{\gamma }}=y_0$, which is exponentially close to the unstable direction of $y_0$. Under $n_{\gamma }$ iterations, the successive images of $y_0$ and $h_{-n^{+}}^{\left(\gamma \gamma \right)}$ first approach, then separate from each other, making a near fly-by somewhere in the middle.

Figure 3

Markov partitions constructed in the Hénon map. Upper panel: The $V_{s_0}$ and $H_{s_{-1}}$ regions corresponds to the same regions in Fig. 1. The four cells $H_{s_{-1}}\cap V_{s_0}\Rightarrow s_{-1}.s_0$ are the Markov partitions of lengths 2. Lower panel: Markov partitions of length 4. The horizontal and vertical strips are created as $H_{s_{-2}{s}_{-1}}=M\left(H_{s_{-2}}\right)\cap H_{s_{-1}}$ and $V_{s_0{s}_1}=V_{s_0}\cap M^{-1}\left(V_{s_1}\right)$. The $H$ and $V$ strips intersect at 16 cells $H_{s_{-2}{s}_{-1}}\cap V_{s_0{s}_1}\Rightarrow s_{-2}{s}_{-1}.s_0{s}_1$, as indicated by a black dot inside each of them. For the sake of clarity, we only explicitly labeled four cells in the lower left corner. Any point from $\mathrm{\Omega }$ with symbolic string of fixed central block $\cdots s_{-2}{s}_{-1}.s_0{s}_1\cdots$ must either locate inside or on the boundary of the $s_{-2}{s}_{-1}.s_0{s}_1$ cell. The sizes of the cells shrink exponentially with increasing string lengths.

Figure 4

(Schematic) The widths of $H_{s_{-n}\cdots s_{-1}}$ and $V_{s_0\cdots s_{n-1}}$ are $\sim O\left(e^{-n\mu }\right)$, so the cell area of $s_{-n}\cdots s_{-1}.s_0\cdots s_{n-1}$ is $\sim O\left(e^{-2n\mu }\right)$.

Figure 5

$a$ and $b$ are arbitrary points and $c$ is a curve connecting them. $a^{\prime }=M\left(a\right),b^{\prime }=M\left(b\right)$ and $c^{\prime }=M\left(c\right)$. Then $A^{\prime }-A=F(q_b,q_{b^{\prime }})-F(q_a,q_{a^{\prime }})$.

Figure 6

$a_0$ and $b_0$ is a homoclinic pair. They are connected by an unstable segment $U[a_0,b_0]$ (solid) and a stable segment $S[b_0,a_0]$ (dashed). Then the action difference between the homoclinic orbit pair is $\mathrm{\Delta }{\mathcal{F}}_{\left\{b_0\right\}\left\{a_0\right\}}=A$.