Floquet-theory for periodic solutions, there is always one trivial multiplier $\mu=1$ or trivial exponent $\lambda=0$, arising from the continuous symmetry with respect to timeshifts in autonomous systems (phase shift on the limit cycle). For a perturbation with $\omega=0$ and $q_{-}=0$ one obtains $$Y_{\pm}(\omega=0,q_{-}=0)=1+\frac{a^{2}}{R}\cos{k_{\tau}}\left(1\pm\frac{\cos{k_{\tau}}}{|\cos{k_{\tau}}|}\right).$$ The corresponding characteristic exponent follows as $$\begin{aligned}
\begin{aligned}\gamma_{+}(\omega=0,q_{-}=0)=\begin{cases}
-\log{\left(1+2\frac{a^{2}}{R}\cos{k_{\tau}}\right)}<0 & \mathrm{for}\,\,\cos{k_{\tau}\geq0}\\
0 & \mathrm{for}\,\,\cos{k_{\tau}<0}
\end{cases}\\
\gamma_{-}(\omega=0,q_{-}=0)=\begin{cases}
0 & \mathrm{for}\,\,\cos{k_{\tau}\geq0}\\
-\log{\left(1+2\frac{a^{2}}{R}\cos{k_{\tau}}\right)}>0 & \mathrm{for}\,\,\cos{k_{\tau}<0}
\end{cases}
\end{aligned}
\label{eq:SL2-torustrivialmultiplier}\end{aligned}$$ Note that this property of the spectrum is not affected by the long delay approximation, since the approximation becomes exact at $\gamma=0$. Apparently there are two parameter domains separated by $\cos{k_{\tau}}=0$. Using (\[eq: SLpsolRe\]), one finds that this boundary corresponds to the curve $a_{U}:=\sqrt{\alpha}$. Thus, all periodic solutions with the amplitudes smaller than $a_{U}$ for a given $\alpha$ are unstable due to a positive characteristic exponent with $\omega=0$. According to [@Cross1993] this instability is called a *uniform instability*. In order to determine the neutral stability curve, the following discussion is restricted to the regime with $\cos{k_{\tau}}\geq0$. Since the relation (\[eq: SL 2-torus gamma shift property\]) holds, we will focus the analysis on $\gamma_{-}(\omega,q_{-})$.

![Asymptotic continuous spectrum $\gamma_{\mathrm{max}}(\omega,q_{-})$ of periodic solutions for different values of $k_{-}$ and $k_{\tau}$. The density plot shows the 