Memory-induced oscillations in a nonlinear GDP–interest rate–debt system with gamma-distributed delays

This paper studies memory-induced oscillations in a nonlinear GDP–interest rate–debt system with delayed foreign-investment feedback. The point delay of a benchmark macroeconomic model is replaced by weak and strong gamma-distributed memories with the same mean implementation time but different temporal shapes. Through the linear-chain technique, the resulting integro-differential system is transformed exactly into four- and five-dimensional autonomous systems, making the spectral and bifurcation structure directly computable. We prove equilibrium inheritance, derive closed characteristic equations, establish stability for small mean delays, and formulate verifiable Hopf and transversality conditions. Under the benchmark calibration, both gamma memories generate transversal Hopf bifurcations at substantially smaller mean delays than the corresponding discrete-delay threshold. Numerical simulations, phase portraits, spectral-abscissa curves, and an amplitude diagram show the transition from damped adjustment to numerically observed attracting periodic regimes. The results indicate that the temporal distribution of memory, and not only its mean, can act as a nonlinear destabilizing mechanism in feedback systems with economic interpretation.