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

## TL;DR

This paper investigates the creation mechanisms of Devil's Staircase surfaces and the coexistence of stable and unstable periodic orbits in the anisotropic Kepler problem, revealing bifurcation phenomena and classifying orbit types.

## Contribution

It introduces a novel tiling mechanism for level surfaces in AKP, classifies periodic orbits based on topology and symmetry, and applies these findings to bifurcation analysis and orbit stability.

## Key findings

- Proper tiling of initial value domain by base ribbons is established.
- Bifurcation from unstable to stable and non-retracing orbits is characterized.
- Gutzwiller's action formula is validated for over 13,648 POs at rank 10.

## 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.