Three-Node Shared-Edge Network under Double-Ended Correlated Load

Abstract

We study a three-node network with six independent directed edges and double-ended correlated load. Under node-permutation symmetry S_3, the twelve-dimensional state space decomposes as \mathbb{R}^{12}=3\cdot\mathbf{1}\oplus\mathrm{sgn}\oplus 4\cdot V_{\mathrm{std}}. The trivial sector reduces to one strictly negative scalar and one two-dimensional coupled block. In the non-orthogonal uniform-edge coordinate \delta w the trivial block is M_0^{\rm draft}; in the orthogonal coordinate compatible with the modal transformation T it is M_0^{\rm orth}, related by the similarity M_0^{\rm orth}=D_0M_0^{\rm draft}D_0^{-1},\qquad D_0=\operatorname{diag}(1,1,\sqrt2), with common characteristic polynomial z^2+(r-s)z-rs-4\lambda u. Hurwitz conditions \ell_1-\ell_2<0,\qquad r>s,\qquad rs+4\lambda u<0, where s=\tfrac{\delta}{2}+4w^*(\chi-\psi), r=2\rho w^*, u=w^*(\alpha+\beta). The sign sector supports an exact one-dimensional equilibrium family; the zero eigenvalue is forced by that family, and the S_3 decomposition identifies the tangent vector as lying in the sign sector. Under isotropic diffusion \eta_A=\eta_B=\eta_{\rm diff}, the standard sector in the non-orthogonal (x,y) coordinate gives the exact orthogonal reduction QM_{\rm std}^{(4)}Q^T=t_{\rm anti}\oplus M_3, with t_{\rm anti}=\ell_1-\ell_2-3\eta_{\rm diff}w^* and M_3=\begin{pmatrix} s_{\rm sym}&-\lambda/\sqrt2&-\lambda/\sqrt2\\ -\sqrt2w^*(\alpha+\zeta)&-2\rho w^*&\rho w^*\\ -\sqrt2w^*(\beta-\zeta)&0&0 \end{pmatrix}, \qquad s_{\rm sym}=\tfrac{\delta}{2}-3\eta_{\rm diff}w^*-2w^*(\chi+2\psi). The characteristic polynomial is P_3(z)=z^3+(r-s_{\rm sym})z^2-(rs_{\rm sym}+\lambda u)z-\tfrac32\lambda rw^*(\beta-\zeta), obtained by direct cofactor expansion of \det(zI_3-M_3); the coefficient -\tfrac32\lambda rw^*(\beta-\zeta) matches the value 3/40 at the admissible parameter point. At the admissible parameter point with \gamma=2/15>0, numerical evaluation of TJT^T confirms that isotypic cross-blocks vanish, that the trivial and sign blocks take their analytic forms, and that the full characteristic polynomial is z\bigl(z+\tfrac{13}{6}\bigr)\bigl(z^2+\tfrac{397}{10}z+\tfrac{39}{10}\bigr)\bigl(z+\tfrac{37}{15}\bigr)^2\bigl(z^3+40z^2+\tfrac{174}{25}z+\tfrac{3}{40}\bigr)^2, with algebraic multiplicity of zero exactly one. For the four copies of the standard representation, the intertwiners U_A=U_B=I_2, U_s, and U_a are constructed by solving the intertwining equations for the generators (12) and (123). Under the standard cycle identification, \rho_a(r)=\rho_{\rm ref}(r) and the two valid U_a matrices differ by an overall sign, U_a'=-U_a. The aligned form J_{\rm aligned}=I_2\otimes M_{\rm mult} is verified numerically at the parameter point; the elementwise similarity M_{\rm std}^{(4)}=SM_{\rm mult}S^{-1} is stated as the remaining open computation. A conditional nonlinear statement (Proposition 7.1) is proved under (H1)–(H4): F\in C^2, local smoothness of the equilibrium family, algebraic multiplicity one at zero, and strict transverse spectral gap at the base point. Then locally the family is a center manifold; the reduced dynamics satisfies \dot a=0 identically; the family is locally transversely attracting; no single equilibrium is asymptotically stable as an isolated point. The phase shift a_\infty-a(0) is not quantified. All results are linear spectral analysis, local nonlinear center-manifold analysis, or one-point numerical evaluation.