# Chaotic Communication System with Symmetry-Based Modulation
## 3 Experimental Results
### Comparing the SCM Approach with Parameter-Based Modulation  
To compare the proposed method of chaotic modulation with existing solutions, one can use bifurcation diagrams and bifurcation spectrograms. The concept of a bifurcation diagram includes several ways of plotting a phase variable on one axis and a parameter on another. One of the simplest ways to plot the bifurcation diagram using the time series is to use a peak detector [23].  
The bifurcation spectrogram is calculated using the windowed Fourier transform. The signal \(s(t)\) is divided into parts, which usually overlap, and then a Fourier transform is performed to calculate the power spectral density \(P\) for each part:  
\[\begin{array}{l}S(t)=FFT(s(t)),\\ P=20\log_{20}(|S(t)|^{2}).\end{array} \tag{18}\]  
The obtained spectra are plotted in one image. Each vertical line corresponds to a single spectrum. Unlike the widely used time-based spectrogram, the bifurcation spectrogram is plotted not depending on time, but depending on the value of the certain system parameter. The bifurcation parameter or the symmetry coefficient may serve as a parameter plotted along the \(X\)-axis of the bifurcation spectrogram.  
The above-mentioned analysis techniques allow investigating one of the main disadvantages of communication systems based on parameter modulation for secure and covert data transmission, namely, the conspicuousness of the parameter switching process. Such switching can be detected by an interceptor while analyzing the signal spectrum. In contrast, modulation based on the symmetry coefficient switching does not affect the behavior of the system. The bifurcation and spectral diagrams of the parameters and the symmetry coefficient of the Rossler system are shown in the Figures 8a-d and 9a-d respectively. Note that symmetry coefficient modulation almost does not affect the system dynamics in comparison with system parameters modulation as it was theoretically predicted.  
The comparison of steps from \(h=0.05\) s to \(h=0.001\) s is given in Figure 10. The smaller is the integration step, the smaller is the influence of the symmetry coefficient on the system dynamics. While decreasing the discretization step, the discrete system becomes closer to the continuous prototype, thus the specific values of the numerical integration method parameters lose their influence on the solution.  
Figure 7: Dependence of synchronization time from the threshold synchronization error \(\Delta x\).  
Figure 8: Bifurcation diagrams for Rössler system simulated with \(h=0.01\) s.  
Figure 9: Spectrum diagrams for Rössler system simulated with \(h=0.01\) s.  
The revealed properties make data transmission by varying the symmetry coefficient much more secretive than transmission by modulating the bifurcation parameter. An illustrative example of transmitting the message \"1010110\" is shown in Figure 11. When a message is transmitted by modulating a parameter, specific patterns can be seen on the spectrogram. Meanwhile, the symmetry-based modulation allows the transmitted symbols to remain undistinguished. One can see, for symbol 1 there are areas where the most marked frequency 0.2-0.6 Hz, which is not the case when transmitting a symbol 0, during the message transmission using parameter modulation. However, such areas are not observed during the message transmission using symmetry coefficient modulation.  
Figure 11: Spectrograms for message transmission using parameter modulation (**a**) and symmetry coefficient modulation (**b**). For parameter modulation, values \(a_{1}=0.2\) and \(a_{2}=0.22\) were switched while values of \(b=0.2\) and \(c=5.7\) were constant. For SCM case, symmetry coefficients were \(S_{1}=0.4\) and \(S_{2}=0.6\), respectively.  
Figure 10: Bifurcation diagrams of the Rössler system for the symmetry coefficient variation at different numerical integration steps.