# Coherent Chaotic Communication Using Generalized Runge-Kutta Method
## 3 Investigation of Chaotic Communication System with \(\alpha\)-Based Signal Modulation  
Let us consider a chaotic communication system based on the discrete Gokyildirim et al. system, obtained by the generalized explicit second-order Runge-Kutta solver, and with a new signal modulation method. The concept of using the numerical integration operator as a tool for modulating the chaotic signal generated by the corresponding discrete map is a recent development in CCS theory. In this section, we describe the principles of the  
Figure 8: The 2D bifurcation diagrams for different values of \(b\) for \(\alpha\) and parameter \(a\).  
construction of a CCS based on the RK-alpha method. We present finite difference schemes for the receiver and the transmitter, and we describe a method for the evaluation of the covertness of the CCS under study. The principle of the chaos-based communication system can be described as follows. At the beginning, the digital information signal \(m(t)\) modulates the RK-alpha value of the chaotic oscillator at the transmitter side. The signal then passes through the communication channel. White Gaussian noise is added in combination with the noise from the DAC and ADC units. We used 12-bit resolution ADC/DAC in our experiments. On the receiver side, depending on the transmitted message bit, the occurrence of generalized chaotic synchronization can be observed. The recovery of the message \(M^{*}(t)\) is done by using the root mean square (RMS). In our previous work [49], we found that using RMS leads to the lowest BER values. A block diagram of the variable center point signal modulation communication system is shown in Figure 9.  
In this study, we use the well-known Pecora-Carroll method [14] to synchronize chaotic oscillators. We add Pecora-Carroll synchronization to the model and obtain the following finite-difference scheme for the slave (receiver) oscillator:  
\[\begin{split}\dot{x}&=y+xy-yz;\\ \dot{y}&=x+ayz+k(y_{M}-y);\\ \dot{z}&=by^{2}+xy;\end{split} \tag{11}\]  
where \(k\) is the coupling strength coefficient, and \(y_{M}\) is the second variable of the master system.  
The resulting receiver system, obtained by the RK-alpha method \(5\), is as follows:  
\[\begin{split} h_{\alpha}&=\alpha*h;\\ x_{n+\alpha}&=x_{n}+h_{\alpha}(y_{n}+x_{n}y_{n}-y_ {n}z_{n});\\ y_{n+\alpha}&=y_{n}+h_{\alpha}(x_{n}+ay_{n}z_{n}+k(y _{M}-y_{n}));\\ z_{n+\alpha}&=z_{n}+h_{\alpha}(by_{n}^{2}+x_{n}y_ {n});\\ x_{n+1}&=x_{n}+h(\alpha_{1}*(y_{n}+x_{n}y_{n}-y_ {n}z_{n})+\alpha_{2}*(y_{n+\alpha}+x_{n+\alpha}y_{n+\alpha}-y_{n+\alpha}z_{n+ \alpha}));\\ y_{n+1}&=x_{n}+h(\alpha_{1}*(x_{n}+ay_{n}z_{n})+ \alpha_{2}*(x_{n+\alpha}+ay_{n+\alpha}z_{n+\alpha})+k(y_{M}-y_{n}));\\ z_{n+1}&=x_{n}+h(\alpha_{1}*(by_{n}^{2}+x_{n}y_{n}) +\alpha_{2}*(by_{n+\alpha}^{2}+x_{n+\alpha}y_{n+\alpha})).\end{split} \tag{12}\]  
Using the approach reported in [47; 49], we have determined that variable \(Y\) is the most optimal for synchronization and has a nearly optimal synchronization coefficient of \(k=4\).  
The example of a test message (1010110010) transmission by the considered chaotic communication system with RK-alpha modulation is shown in Figure 10.  
Figure 9: Block diagram of the communication system with RK-alpha.