A Mexican hat dance: clustering in Ricker-potential particle systems

The paper linearizes the gradient-flow ODE at the fully-stacked origin, computes the Jacobian entries Jjj|O = -2/w^2 + (n-1)·2k^2/((k-1)s^2) and Jji|O = -2k^2/((k-1)s^2) for i≠j, and identifies the eigenvalues λ1 = -2/w^2 (once) and λ⊥ = -2/w^2 + 2nk^2/((k-1)s^2) (multiplicity n-1). This yields the stability threshold w0 = (s/k)√((k-1)/n) for the origin state, exactly as stated in the paper’s Stability subsection (equations (10)–(12) following the system definition and derivative formulas) and consistent with the figure captions defining w0 as the critical value where the origin becomes unstable . The candidate solution reproduces the same result via an equivalent matrix form, J = -(2/w^2)I + A(nI - 11^T) with A = 2k^2/((k-1)s^2), recognizing the complete-graph Laplacian spectrum and deriving the same eigenvalues and threshold. Hence both are correct and essentially the same linearization argument.