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Use the same method of multiple scales and its solvability condition in the method of multiple scales. The position-dependent damping function is odd, so the lemma that odd position-dependent damping has zero first-order amplitude drift applies. In its wave amplitude average, $\psi \mapsto \psi +\pi$ changes $\sin (A\cos \psi )$ to its negative while preserving ${\sin }^2\psi$, so the average is zero. Hence $A^{\prime }=0$; the wave phase average again gives ${\theta }^{\prime }=0$.

Equivalently, expand the position-dependent damping in its uniformly convergent power series for bounded $A$. Each term ${\cos }^{2m+1}\psi \sin \psi$ has only even sine Fourier harmonics, so none resonates with the unit-frequency oscillator. With the initial wave amplitude and wave phase unchanged, the leading uniform approximation isThere is no first-order slow drift. Effects at the next order accumulate only an $O(ϵ)$ correction on this time scale.

In the weak, non-crossing, leading-eccentricity model the exterior resonant coefficient has positive sign: write its disturbing function as $R_{\mathrm{res}}=B_q{e}_c^q\cos {\varphi }_{bc}$, with $B_q>0$ for the orders considered. The physical feedback can be seen for first order. Before an outer conjunction near apoapsis, the faster inner planet pulls the outer one backwards; after passing it pulls forwards. On the outward side, the nearer, stronger earlier pull wins, reducing the outer semi-major axis and increasing its mean motion. Thus ${\varphi }_{bc}<\pi$ is driven upward. On the inward side, the later forward pull wins and drives ${\varphi }_{bc}>\pi$ downward. The paired impulses cancel at the symmetric centre.

For higher order, sum the feedback over all $q$ conjunction phases. The first surviving angular Fourier harmonic is $\sin {\varphi }_{bc}$; the lower Fourier harmonics cancel over the equally spaced directions. A local Hamiltonian reduction with canonical momentum $J$ hasThe negative curvature of the Kepler orbit Hamiltonian gives the displayed sign. Linear stability analysis at $\pi$ gives a restoring acceleration. The symmetric libration centres of an eccentricity resonance are thereforeThe corresponding conjunction phases relative to periapsis are $\{\pi \}$, $\{\pi /2,3\pi /2\}$ and $\{\pi /3,\pi ,5\pi /3\}$. They suppress repeatedly close passages and provide resonance protection.

These are expected centres of the specified leading-harmonic model, not centres fixed solely by the integer order for arbitrary orbital eccentricity and masses. Asymmetric resonant-argument libration can occur when higher Fourier harmonics matter. For example, if $R=A\cos \varphi +B\cos 2\varphi$, $A>0$, $B>A/4$, the symmetric $\pi$ equilibrium loses stability and stable centres satisfy $\cos \varphi =-A/(4B)$. Thus further dynamical information is needed beyond the PDF's unrestricted eccentric orbit to assert unique centres. Exterior asymmetric branches are documented by Winter and Murray, “Resonance and chaos. II. Exterior resonances and asymmetric libration”.

Now the eccentric planet $c$ is inside the circular perturber $d$. The interior eccentricity-type mean-motion resonance hasIn the mean-conjunction approximation, ${\varphi }_{cd}=n(\Lambda -{\varpi }_c)$, so ${\Lambda }_j={\varpi }_c+({\varphi }_{cd}+2\pi j)/n$. Successive conjunctions advance by $2\pi m/n$; coprime $m,n$ visit $n$ distinct phases.

The near-conjunction tangential pulls are reversed for an eccentric body inside its perturber. Just before conjunction the inner body is pulled forwards and afterwards backwards. Near periapsis, an inward-moving inner body experiences the stronger earlier forward pull; its semi-major axis grows, its mean motion falls, and ${\varphi }_{cd}$ increases because its coefficient of ${\lambda }_c$ is negative. On the outward side the feedback reverses, restoring the first-order centre zero. Higher-order feedback is strongest on the side closest to the external perturber, near apoapsis; the leading coefficient alternates in sign with order.

Writing $R_{\mathrm{res}}=B_n{e}_c^n\cos {\varphi }_{cd}$ gives $B_n<0$ for odd $n$ and $B_n>0$ for even $n$ in the leading non-crossing model. The same negative-curvature Hamiltonian argument therefore givesConjunction phases relative to periapsis are $\{0\}$, $\{\pi /2,3\pi /2\}$ and $\{0,2\pi /3,4\pi /3\}$, respectively, as shown in the lower row of the figure. As before, these are small-eccentricity symmetric predictions; additional resonant Fourier harmonics and coupled secular dynamics can change the stable branches.

For actual alignment of the eccentric body at true longitude $\Lambda$, the exact phase relation is instead ${\varphi }_{cd}=(m+n)x-my$, where $x=\Lambda -{\varpi }_c$ and $y=M_c(x)$. The simple equally spaced angles replace $y$ by $x$ at leading order.