Algebraic Resonance and Nonlinear Hopf Structure in Circulant Networks: A Complete Classification and its Psychological Interpretation

We present a complete analytical classification of Hopf bifurcations in C_N circulant networks of the form: dxᵢ/dt = α xᵢ - g₃ xᵢ³ + κ xᵢ₊₁ - μ xᵢ₊d with indices modulo N. We derive closed-form expressions for the spectrum, the Hopf threshold, and the nonlinear normal form coefficients of arbitrary odd order. We prove that the cubic coefficient is universal and scales as -3g₃/N, while higher-order coefficients follow the exact formula: a₂ₚ₋₁ = - [(2p-1)! / 2] g₂ₚ₋₁ N¹⁻ᵖ We identify an arithmetic resonance structure governed by modular congruences, leading to a classification in terms of gcd(N, 15). We show that C₅ is algebraically singular in this hierarchy. Finally, we provide a structural interpretation of each topology in terms of Jungian individuation dynamics.We add several python codes (in spanish) as complementary work.