Resonant and non-resonant Double Hopf bifurcation in a 4D Goodwin model with Wage inequality

This paper presents a four-dimensional extension of the Goodwin model of endogenous cycles that integrates wage inequality and underemployment. The model distinguishes two classes of workers differentiated by productivity, wage levels, and bargaining strength, and endogenizes the underemployment rate through a simplified power-balance mechanism between capital and labor. We establish well-posedness of the system by proving existence–uniqueness of solutions, positivity, and forward invariance on a compact admissible set. The interior equilibrium is characterized in closed form and shown to generically undergo a double Hopf (Hopf–Hopf) bifurcation. Using center–manifold reduction and a third-order normal form, we derive the amplitude equations governing the interaction between two oscillatory modes (the Goodwin cycle and the underemployment cycle). The reduced dynamics predict the emergence of an invariant two-torus with quasi-periodic cycles and phase locking at low-order resonances ($1{:}1$, $1{:}2$, $1{:}3$). Numerical continuation and direct simulations corroborate the analytical predictions, documenting transitions between quasi-periodicity and resonant periodic orbits, and mapping the associated bifurcation structure in key parameters, such as the adjustment speed of the underemployment rate in response to deviations from steady-state equilibrium.