Stability switching and Hopf bifurcation in a Keen model with gamma-distributed Phillips-curve memory

This paper revisits the delayed Keen model with inflation by replacing the point delay in the Phillips-curve channel with weak- and strong-gamma memories. The linear-chain trick converts the two distributed-memory formulations into four- and five-dimensional autonomous systems with polynomial characteristic equations. For the calibration inherited from the discrete-delay reference model, an exhaustive imaginary-root search reveals a qualitative distinction that is missed by a short-memory analysis. The weak-gamma equilibrium is stable for 0 < T < 0.254861, unstable for 0.254861 < T < 64.636669, and stable again for T > 64.636669. The strong-gamma equilibrium loses stability at T=0.114032 (mean memory 0.228064) and has no additional positive imaginary-root crossing. First Lyapunov coefficients at the two onset points are negative, establishing supercritical Hopf bifurcations with locally attracting periodic solutions. A second positive equilibrium is also identified, but it has a positive real eigenvalue for every T > 0 and is therefore excluded from the stability-switch analysis. These results show that the first destabilization mechanism persists under both gamma kernels, whereas the global dependence on memory length is kernel-specific: exponential memory permits re-stabilization, while the order-two Erlang memory does not for the parameter set studied.