Creation Mechanism of Devil's Staircase Surface and Unstable and Stable Periodic Orbit in the Anisotropic Kepler Problem

Abstract

A long-standing question in two dimensional Anisotropic Kepler Problem (AKP) concerns with the uniqueness of an unstable periodic orbit (PO) for a given binary code (modulo symmetry equivalence). In this paper, a finite level ($N$) surface defined by the binary coding of the orbit is considered over the initial value domain $D_0$. It is proved that a tiling of $D_0$ by base ribbons of the surface steps is proper; the surface height increases monotonously when ribbons are traversed from left to right. The mechanism of level $N+1$ tiling creation from $N$ one is clarified. Two cases are possible depending on the code and the anisotropy. (A) Every ribbon shrinks to a line at $N \rightarrow \infty$. Here the uniqueness holds. (B) When future (F) and past (P) ribbon become tangent each other, they escape from shrinking, Then, the initial values of a stable PO ($S$) and an unstable PO ($U$) sharing the same code co-exist inside the overlap of F and P non-shrinking ribbons. This case corresponds to Broucke's PO. At high anisotropy, it is only case (A), but with decreasing anisotropy, bifurcation $U(R) \rightarrow S(R) +U'(NR)$ occurs, along with the emergence of a non-shrinking ribbon. (Here $R$ and $NR$ are short for self-retracing and non-retracing PO respectively). We conjecture that case (B) occurs only for odd rank, $Y$-symmetric POs from a classification based on topology and symmetry. We report two applications. First, the classification is applied successfully to the successive bifurcation (above bifurcation is followed by $S(R) \rightarrow S'(R) +S''(NR)$) of a high-rank PO ($n=15$). Second, enhancing sensitivity to co-existence of S and U POs by ribbon tiling, we examine high anisotropy region. A new symmetry type PO ($O$ type) is found and, at $\gamma =0.2$, all POs are unstable and unique. 13648 POs at rank 10 verifies that Gutzwiller's action formula amazingly works.