Nonlinear stability conjecture for N-spot ring patterns

Let an $N$-spot ring pattern have spot-spot distance equal to a stable zero $d_c$ of the interaction function $f$, so that $f(d_c)=0$ and $f'(d_c)>0$. The distance $d_c$ is called a binding radius, and the smallest such distance is the smallest binding distance. The $N$ zero eigenvalues associated with neutral modes are excluded from consideration.

Nonlinear stability conjecture. All $N$-spot ring patterns are nonlinearly stable unless the spot-spot distance is the smallest binding distance.

The preceding linear analysis gives eigenvalues with non-positive real parts after excluding the neutral modes; the conjecture concerns the effects of higher-order terms and asserts nonlinear stability for every binding radius except the smallest one.