Transitions and chaos of a business cycle model

This study investigates a modified Bouali model (referred to as the business cycle model) through the lens of transition dynamics theory. Under reasonable theoretical assumptions, we identify the output-capital ratio $c$ as the key parameter that determines the transition and chaos in the business cycle model. Through rigorous mathematical analysis, we demonstrate that this model exhibits both jump (discontinuous) and continuous transitions as the output-capital ratio $c$ increases and crosses a critical value. In particular, for the output-capital ratio large enough $c$, the business cycle model develops a strange attractor, which means that there exists chaos in this business cycle model.