= Fixed points of a damped first-order exterior resonance
{title2=$e_f^2=2/[5(p+1)]$}
For positive locally constant $A,C$, supplement <damped first-order resonant complex eccentricity> by $\dot a/a=-2(p+1)Ce\sin\phi-4A/5$. Balancing the two dissipative rates gives $e_f^2=2/[5(p+1)]$, $\sin\phi_f=-Ae_f/C$ and $Bx_f=-(C/e_f)\cos\phi_f$. Two fixed points exist when $A<C\sqrt{5(p+1)/2}$. The negative-cosine branch has $x_f>0$ and is stable in this leading model: a phase increase raises the positive resonant <torque>, raises <semi-major axis>, lowers <mean motion>, and reverses the phase drift. The positive-cosine branch has the opposite feedback.
For a coupled verification put $z=u+iv$. The <characteristic polynomial> at either fixed point is $\lambda^3+2A\lambda^2+[A^2+B^2x_f^2-2B(p+1)Cu_f]\lambda-4AB(p+1)Cu_f$. For $u_f<0$ its positive coefficients satisfy the cubic <Routh-Hurwitz stability criterion>, since the remaining inequality reduces to $2A(A^2+B^2x_f^2)>0$. For $u_f>0$ its negative constant term forces a positive real root. At the equality threshold the equilibria merge and an <eigenvalue> is zero. This is local stability for the frozen smooth-coefficient approximation, not a conclusion for arbitrary rapidly varying dissipation laws.
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