Chaotic economic systems can be viewed as unpredictable due to their sensitivity to initial values, which makes forecasting extremely difficult [15, 21, 41, 74]. This is known also as the butterfly effect [57]. Devaney [30] proposed that sensitivity in conjunction with other properties, namely transitivity and density of periodic solutions , be considered as ingredients of chaos. An alternative way to prove the presence of chaos is by observing the period-doubling cascade [38]. This chaos is also sensitive, since there are infinitely many solutions with different periods and they are unstable . We utilize these ways of observing chaos in our paper. Importantly, irregularity based on theoretical deterministic chaos can be visualized in simulations. 

One should remark that it is not only sensitivity that can be considered as a mathematical represen- tation of unpredictability, but also the existence of infinitely many unstable periodic solutions. Indeed, while the presence of a single periodic solution can be accepted as a strong indicator of predictability (if one knows the values of the process during the period, then one knows all its future values), with infinitely many unstable periodic solutions all the cycles are unstable, and the trajectory of the dynamics wanders around, visiting neighborhoods of the cycles in an unpredictable way. That is the reason why in the literature the proof of the existence of a periodic-doubling cascade is accepted as evidence of chaos. Stabilizing periodic solutions is named in chaos theory as control of chaos. 

Chaos theory could provide a new approach to economic policy-making. Economists believed initially that chaotic dynamics is not only unpredictable, but also un-controllable. The results of Ott et al. [67] showed that control of a chaos can be made by a very small corrections of parameters [39, 46]. This and related methods have been widely applied to economic models, as exemplified by Holyst et al. [47], Kaas [49], Mendes and Mendes [59], Chen and Chen [24] and many others. 

In the classic book [76] it is observed that while forced oscillator systems naturally emerge in theo- retical investigations of several technical and physical devices, economic examples for this special family of functions have only rarely been provided. The main reason for this deficiency may lie in the fact that the necessary periodicity of the dynamic forcing may not be obvious in most economic applications. Our proposals are to apply deterministic and chaotic exogenous shocks to economic models and make them more realistic. 

One may view chaos (the lack of forecasting) as undesirable in economics, but unavoidable. Hence a deterministic economic model is realistic if it exhibits chaotic motions. We suggest considering the presence of chaos in a model not only as an indication of its adequacy, but also as a measure of its power. Indeed, the presence of chaos implies that the model generates infinitely many aperiodic motions and motions with different periods, which are unstable, and consequently easily affected by control and sustained in a desirable mode. In other words, deterministic chaos is essential for the flexibility and high-speed adjustment of economic models, an indispensable feature in the modern world. 

The principal novelty of our investigation is that we create a chaotic perturbation, plug it in a regular 

