# Periodic orbits in a near Yang-Mills potential

## TL;DR

This paper investigates the stability and bifurcation patterns of periodic orbits in the Yang-Mills potential and related potentials, revealing that only certain period-9 orbits are stable.

## Contribution

It introduces the analysis of bifurcations and stability of periodic orbits in the Yang-Mills potential, highlighting the unique stability of period-9 orbits.

## Key findings

- Period-9 orbits are stable in the YM potential.
- Bifurcation patterns are consistent across symmetric odd-period orbits.
- All orbits are unstable at the YM potential (α=0).

## Abstract

We consider the orbits in the Yang-Mills (YM) potential V=1/2 x2 y2 and in the potentials of the general form Vg=1/2 [{\alpha} (x2 +y2)+x2 y2]. The stable period-9 (number of intersection with the x-axis, with ) orbit found in the YM potential is a bifurcation of a basic period-9 orbit of the Vg potential for a value of {\alpha} slightly above zero. This basic period-9 family and its bifurcations exist only up to a maximum value of {\alpha}={\alpha}max. We calculate the Henon stability index of these orbits. The pattern of the stability diagram is the same for all the symmetric orbits of odd periods 3,5,7,9 and 11. We also found the stability diagrams for asymmetric orbits of period 2,3,4,5 which have again the same pattern. All these orbits are unstable for {\alpha}=0 (YM potential). These new results indicate that in the YM potential the only stable orbits are those of period-9 and some orbits with multiples of 9 periods.