## 3 Colpitts Chaos Oscillator
### Study on the Colpitts Chaos Oscillator in a Simulation Environment and Prototype
#### 3.3.1 Simulation Study  
Following modifications (\(L_{1}\) = 10 uH and \(C_{1}\) = \(C_{2}\) = 5.4 nF), the Colpitts chaotic oscillator model was created in the LTspice simulation environment. Resistor nominal and supply voltage values were used the same as in the original study [36], as well as the 2N2222 NPN bipolar junction transistor. The simulation lasted 500 us and used a maximum timestep of 0.1 us. Signals generated in the LTspice simulation environment were exported to text files and processed in MATLAB. Following the simulation, the two-dimensional projections of the resulting attractor, as shown in Figure 4, were compared to those from the original study [36]. The observed congruence between the projections indicates that the LTspice model was successfully created. The Z1 TEST result was 0.99, suggesting that the system's behavior is chaotic.  
The next goal was to synchronize two oscillators in the drive-response system. The model of the Colpitts chaos oscillator in the LTspice simulation environment was expanded--the second identical Colpitts chaos oscillator circuit was added. The synchronization was established through an operational amplifier in voltage-follower mode. After the simulation, signals were exported and processed in MATLAB. The correlation coefficient between corresponding nodes in the circuit was used to evaluate synchronization. This assessment was used to determine the level of synchronization attained. Research revealed that synchronization via the substitution method in [PERSON]'s chaos oscillator can be established by utilizing voltage across capacitors \(C_{1}\) or \(C_{2}\). The correlation coefficient values in both configurations reached a notable 0.98, proofing the system's reliable synchronization.  
Further insights into the system dynamics were obtained by analyzing signal spectra, contributing to an enhanced understanding of its behavior. Figure 5 presents the \(V_{C1}\), \(V_{C2}\), and \(I_{L1}\) signal spectra. A careful examination of the signal spectra reveals that the signal band is mostly concentrated around the fundamental frequency of the chaos oscillator. It is also worth noting that the signals have a wide band with gradually decreasing amplitudes. This specific signal feature demands careful consideration while calculating filters for the communication system. Understanding these spectral characteristics is critical for enhancing system performance and guaranteeing efficient signal processing.  
Furthermore, cross-correlation functions were calculated within a single Colpitts chaos oscillator to evaluate the chaotic behavior and determine the shortest bit length [59]. Figure 6 depicts the cross-correlation functions between \(V_{C1}\) and \(V_{C2}\), \(V_{C1}\) and \(I_{L1}\), and \(V_{C2}\) and \(I_{L1}\).  
The examination of aperiodic signals (as chaotic signals) indicates a rapid reduction in the amplitude of the cross-correlation function as the time shift between signals grows (Figure 6), indicating the oscillator's complicated and non-repetitive dynamics. This unpredictability is especially useful for applications that require high randomness and complexity. The bit length is calculated by identifying the settling point of the amplitude  
Figure 4: Colpitts Chaos oscillator attractor two-dimensional projections in LISpice simulation.  
Figure 5: Colpitts Chaos oscillator signals \(V_{C1}\) (blue), \(V_{C2}\), (red) and \(I_{L1}\) (yellow) spectra.  
inside the cross-correlation function and linking it with the associated time shift. This method provides the proper choice of minimal bit length [59]. Based on the cross-correlation function analysis, a minimum bit length of 6 \(\upmu\)s is required to ensure its distinguishability during detection, limiting the data transmission rate to 166.7 Kbps. Periodic signals, on the other hand, show a progressive decrease in the amplitude of the cross-correlation function with increasing time shift, indicating predictable and repeated activity inside the oscillator.