# Coherent Chaotic Communication Using Generalized Runge-Kutta Method
## 4 Conclusions  
In this study, we investigated the dynamics of the Gokyildirim chaotic system discretized by the generalized explicit second-order Runge-Kutta integration method. Using the bifurcation diagrams, Lyapunov spectrum, and basin of attraction analysis, we discovered multistable behavior within a certain range of \(\alpha\) values in the discrete Gokyildirim system. Taking this knowledge into account, we developed a novel chaotic signal modulation technique based on changing the value of \(\alpha\) in discrete chaotic maps obtained by the generalized explicit second-order Runge-Kutta solver. This technique has potential applications in various fields of chaotic communications and chaos generation as it is suitable for controlling the properties of generated chaotic sequences. We applied quantified return map analysis and bit error rate analysis to the prototyped chaotic communication systems to evaluate the efficiency of the proposed modulation technique. Our analysis showed that the proposed modulation method is suitable for secure data transmission in chaos-based communication systems. An experimental test bench was designed to investigate and compare the suggested modulation method with state-of-the-art chaos modulation techniques. The test bench includes a transmitter, receiver, and noise modules. The experimental results demonstrate that the proposed \(\alpha\)-based modulation has high noise resistivity and secrecy, especially when compared to the conventional parameter modulation (PM) approach. A chaotic communication system based on \(\alpha\) modulation outperforms its counterpart with parameter modulation by 10-15% in terms of secrecy in all experimental cases and has similar noise immunity. Therefore, it can be concluded that the proposed \(\alpha\)-based modulation technique is a simple and versatile method for constructing secure and noise-resistant transceivers for and receivers of chaotic signals for the implementation of chaos-based secure communication systems. The known limitations of the proposed technique can be formulated as follows. The alpha-RK-method-based modulation can be applied only to chaotic oscillators that are described by ODEs. The applicability of this technique to systems described by algebraic differential equations is questionable, and it is certainly not applicable to systems represented by discrete maps. In addition, the technique is fairly useless for noncoherent chaotic communications. The possible directions for future research include designing chaos generators based on fractional-order or stochastic differential equations.  
Conceptualization, D.B.; data curation, V.A.; formal analysis, V.R., V.A. and T.K.; investigation, I.B. and V.R.; methodology, T.K. and D.B.; project administration, T.K.; software, I.B. and V.R.; supervision, D.B.; validation, I.B., V.R., V.A., T.K. and D.B.; visualization, I.B.; writing--original draft, I.B., V.R. and D.B.; writing--review and editing, V.A. and T.K. All authors have read and agreed to the published version of the manuscript.  
Figure 15: QRMA vs. SNR estimation.  
**Funding:** This study was supported by the Russian Science Foundation (RSF), project 20-79-10334.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Acknowledgments:** The authors are grateful to [PERSON] for his time and fruitful discussions on the topic of this research.  
**Conflicts of Interest:** The authors declare no conflicts of interest.