Baumol and Benhabib [15] summarized the significance of chaos research for economics: “Chaos theory has at least equal power in providing caveats for both the economic analysis and the policy designer. For example, it warns us that apparently random behavior may not be random at all. It demonstrates dramatically the dangers of extrapolation and the difficulties that can beset economic forecasting generally. It provides the basis for the construction of simple models of the behavior of rational agents, showing how even these can yild extremely complex developments. It has served as the basis for models of learning behavior and has been shown to arise naturally in a number of standard equilibrium models. It offers additional insights about the economic source of oscillations in a number of economic models.” 

Indeed, applications of chaos theory have illustrated the possibility of producing complex dynamics in deterministic settings [19, 20, 35, 41, 58, 72, 88], with some papers specifically focusing on building “chaotic business cycles" [32]. Chaos is generated endogenously, and its appearance hinges on the values of some crucial parameters of the model. The main novelty of our paper is that we start with a model that is not endogenously complex. In one case (model$_{A}$$_{4}$ in the main body), we assume that the system has a limit cycle, where the limit cycle is understood to be a closed orbit that is also an attractor [45]. We then subject the model to chaotic exogenous shocks and obtain a perturbed system that admits chaotic motions. The chaos emerging around the original limit cycle is cycle-like, and therefore can be called a chaotic business cycle. This approach is based on rigorous mathematical theory [2, 6], and we provide numerical simulations. In another case (model$_{A}$$_{5}$ in the main body), we subject a system with an asymptotically stable equilibrium to chaotic cyclic shocks, which produces a chaotic business cycle in the original model, as well. We demonstrate this scenario with simulations, as this approach does not have theoretical underpinnings as yet. 

In this paper we show that it is possible to produce a chaotic business cycle in a very natural way - take a system of differential equations with a limit cycle as a point of departure, and introduce a chaotic exogenous disturbance. An example of an exogenous disturbance is a technology shock to the economy which affects output, holding all other variables constant. We describe it using solutions of chaos generator models. We use them to demonstrate the proposed approach, and other formulations can be studied in future work. For example, one can use actual economic time series, such as commodity prices, that have been tested for deterministic chaos [14, 21, 34, 68]. Moreover, shocks other than technology shocks can be considered, in view of the on-going debate between two literatures supporting and rejecting the importance of technology shocks for generating business cycles [12, 37, 54]. 

Our results give more theoretical lights on the processes, as we suggest a mathematical apparatus, which describe rigorously extension of chaos , increases its complexity, and provides new structures of effective control for clusters of economic models. 

