*4.7. Industrial Application Case: Financial Chaotic System*

Due to the nonlinear nature of the financial markets, chaos models using nonlinear dynamics have been a popular topic in recent years. Uncertainty in the market environment has a particularly negative impact on the financial system. Therefore, describing the financial chaos model with random elements is more practical. Due to deterministic instability, financial chaos, such as the extreme turbulence of the financial market and the financial crisis, occurs during the functioning of the financial system, which has significant detrimental effects on economic development and social stability. Controlling the financial system from a chaotic to a periodic state is as simple as modifying the controller settings. As a first step, we theoretically obtained a range of values for the controller parameters by analyzing the financial system's dynamic equations and controllers. Later, we investigated the effects of these parameters on the system.

#### **5. Conclusions**

Chaotic synchronization is key for chaotic signals in a communication system. On the receiver end, the chaotic system's parameters are unknown; thus, the task is to determine the ideal values to retrieve the message signal. Using the fruit fly optimization technique, this article improved chaotic synchronization in chaos-based wireless networks. In this study, parameter estimation for a three-dimensional Lorenz chaotic system was set up as a multi-dimensional optimization problem and solved using the quantum fruit fly optimization method. Quantum theory was employed by the FOA model and replaced the osphresis-based search of FOA with a quantum behavior-based searching mechanism. The quantum fruit fly optimization technique improved parameter estimation accuracy by carefully exploiting the search space and converging, which suggested that the algorithm could estimate optimum parameter values. Furthermore, it enhanced the exploration of optimal solutions by sharing information regarding parameter values. The difference between the proposed model and existing metaheuristic algorithms was the use of fruit fly optimization to produce better quality solutions and convergence speed, i.e., establishing an optimal trade-off between exploration and exploitation. This model may be extended to other chaotic systems.

The results and discussion of this study led to the following conclusions (important results): (1) Numerical simulations indicate the proposed approach can accurately predict chaotic system parameters. The suggested model is faster and more accurate than current techniques. This outcome is due to balancing exploitation and exploration in the search space. (2) Even with the original signal added to the chaotic signal, the current algorithm can still identify it well, especially for the Lorenz system. (3) As with final estimated results, 30 samples of data has the highest accuracy and least variation, proving that the amount of input data affects algorithm stability.

For future work, the proposed model should be applied to different chaotic systems, such as in high-dimensional, hyper chaotic systems, and time-delay chaotic systems. Implementation and testing in a real testbed are important in the field of wireless communication. Real deployment tests can bring up issues that did not come up in simulation. To work well in real implementations, changes to the proposed model may be required.

**Author Contributions:** Conceptualization, S.M.D.; methodology, S.M.D.; software, Q.M.Z. and M.B.K.; validation, S.M.D.; formal analysis, S.M.D. and M.B.K.; investigation, S.M.D., Q.M.Z. and M.B.K. resources, Q.M.Z. and M.B.K.; data curation, S.M.D.; writing—original draft preparation, S.M.D., Q.M.Z. and M.B.K.; writing—review and editing, S.M.D.; visualization, Q.M.Z. and M.B.K.; supervision, S.M.D.; project administration, Q.M.Z. and M.B.K.; funding acquisition, Q.M.Z. and M.B.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** The study did not require ethical approval.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The study did not report any data.

**Conflicts of Interest:** The authors declare no conflict of interest.
