Characteristic multipliers of figure-eight in -1 < \alpha < 2.5

Four non-trivial characteristic multipliers for the figure-eight solution are shown. Here, \alpha is the power of potential energy, U=-\alpha^{-1}/r^{\alpha}.

If you prefar GIF animation, here it is.

Figure-eight in -1 < \alpha has 5 regions of stability, EE <--> Q <--> EE <--> HE <--> HH. So, there are 2 separated EE regions.

Here, E stands for a pair \mu and \mu^{*}(complex conjugate) with |\mu|=1,
H stands for a pair \mu \in \mathbb{R} and 1/\mu,
and Q stands four eigenvalues \mu, \mu^{*}, 1/\mu and 1/\mu^{*} differ from each other. See the figures.