Shrimp-shaped structure and period-bubbling route to chaos in a one-dimensional economic model

A beautiful feature of nature is its complexity. The chaos theory has proved useful in a variety of fields, including physics, chemistry, biology, and economics. In the present article, we explore the complex dynamics of a rather simple one-dimensional economic model in a parameter plane. We find several organized zones of "chaos and non-chaos" and different routes to chaos in this model. The study reveals that even this one-dimensional model can generate intriguing shrimp-shaped structures immersed within the chaotic regime of the parameter plane. We also observe shrimp-induced period-bubbling phenomenon, three times self-similarity of shrimp-shaped structures, and a variety of bistable behaviors. The emergence of shrimp-shaped structures in chaotic regimes can enable us to achieve favorable economic scenarios (periodic) from unfavorable ones (chaotic) by adjusting either one or both of the control parameters over broad regions of these structures. Moreover, our results suggest that depending on the parameters and initial conditions, a company may go bankrupt, or its capital may rise or fall in a regular or irregular manner.