20 October, 2026

Notes

2026-10-30

To do!

tracing surface of section

I have two scans where I search periodic orbits and look random trajectories, in order to see how the structures change for the various energy shells between \(E_1=-0.5\) and \(E_2=-0.3\). This region is particular interesting, as this region is where the structure of phasespac change from partially structured system to a completly chaotic system.

A particular interesting region, is the energy range: \(E=-0.45765\frac{\mathcal{C}}{L}\) and \(E=-0.43882 \frac{\mathcal{C}}{L}\).

I also tested this transition for various gap sizes, though the simulation for \(del=10^{-3}, 10^{-5}, 10^{-8}\) still crashes for the chaotic region. Interestingly the qualitatively, the region inspected (\(E\in[-0.5,-0.3]\)) seems qualitatively unaffected.

Here is one energy picked randomly, to allow for a comparison.

E=-0.391 C/L

(a) gap: 10⁻⁸

gap: 10⁻⁶

(b) gap: 10⁻³

Figure 1: E= 0.4145 C/L

(a) gap: 10⁻⁸

gap: 10⁻⁶

(b) gap: 10⁻³

Figure 2

Lost Wonderers Thoughts: