# Chaotic Communication System with Symmetry-Based Modulation
## 3 Experimental Results  
### SCM Transfer Analysis  
Let us estimate the synchronization time for various initial conditions and determine the optimal symmetry and synchronization coefficients using average and maximum synchronization time. All computer experiments were performed with the extended floating-point data type using NI LabVIEW 2020 and MATLAB 2021a software. The results for two sets of different initial conditions for master and slave systems with size 200 are shown in Figure 4. One can see, the value of the symmetry coefficient has almost no effect on the synchronization time. For obtained results, one can conclude that the optimal value of \(k\) for Rossler system equals 1.4. Consider a case of simple peer-to-peer data transfer based on synchronized discrete Rossler systems with adaptive symmetry. If the binary message is encoded by a two-symbol alphabet, then the transmitter is modulated with two different symmetry coefficients \(S_{1}\) and \(S_{2}\). For example, one can take \(S_{1}\) = 0.4 and \(S_{2}\) = 0.6 to denote logical 1 and 0. In idle state, master oscillates with \(S_{idle}=0.5\). This value is important for the data transfer rate, as will be demonstrated further. To receive the message, a system consisting of two slave oscillators in a peer-to-peer configuration (see Figure 2) is needed. The synchronization error behavior in receiver systems during transmission of the logical 1 is shown in Figure 5.  
Figure 4: The dependence between average (**a**) and maximum (**b**) synchronization time and the symmetry and synchronization coefficients. Figure 5: Normalized synchronization errors at receiver systems during a symbol transmission associated with value \(S_{1}\) at the transmitter. Note that here the Rossler system is considered in its natural time-scale (4). The system was simulated with the time step \(h=0.01\) and required 100 s of simulation time per symbol. With this time step, the maximal synchronization accuracy for the double-precision data type was achieved within \(6\cdot 10^{3}\) samples. An important observation here is that synchronization error \(\Delta x\) decreases exponentially over time. In a real communication system, chaotic oscillators can be executed with an almost arbitrary frequency depending only on a sample rate. If the transmission is carried out with a sampling frequency of 1 GHz, then the minimal synchronization error will be achieved in 6 \(\upmu\)s.  
Let us illustrate the transmission of binary messages using the proposed modulation technique. The synchronization error behavior while transmitting the message \"10101100\" is shown in Figure 6.  
From a practical point of view, achieving the minimum possible synchronization error value during the transmission is unnecessary and even almost impossible due to the noise presented in a real communication channel. Meanwhile, it is sufficient that the synchronization error of one of the receivers responsible for the symbol associated with a specific value of \(S\) should fall below a certain threshold \(\Delta x_{0}\), for example, \(\Delta x_{0}=10^{-6}\). The second practical consideration is that for a case of the dense channel occupation, it is of interest to choose the smallest possible difference \(\Delta\) between the values of the symmetry coefficients associated with the symbols: \(\Delta S=S_{i}-S_{j}\to 0\). An additional advantage of such an approach is that the trajectories of the master and slave systems with slightly different values of the symmetry coefficients do not diverge much, and when a certain symbol is transmitted, the error decreases below the specified threshold \(\Delta=x_{master}-x_{slave}\) faster.