 Asymmetric resonant-argument libration

Libration about a resonant phase distinct from the symmetric values zero and $\pi$. For example, in a negative-Kepler-curvature reduction with disturbing potential $R=A\cos \varphi +B\cos 2\varphi$, $A>0$, $B>A/4$, stationary points obey $\sin \varphi (A+4B\cos \varphi )=0$. At $\pi$, $R^{\prime \prime }=A-4B<0$, so it is unstable, whereas at $\cos \varphi =-A/(4B)$ one has $R^{\prime \prime }=4B-A^2/(4B)>0$, giving two stable asymmetric centres. Actual branch thresholds depend on the resonance coefficients and evolving orbital elements.