## 3 Colpitts Chaos Oscillator
### Chaotic Synchronization  
Establishing chaotic synchronization between chaos oscillators within the transmitter and receiver is crucial to ensuring secure communication in coherent data transmission systems. [PERSON] and [PERSON] introduced a synchronization method [PERSON] and [PERSON] (1983) that involves substituting the signal within the response circuit with a signal sourced from the drive circuit. This approach facilitates information recovery from the received signal, constituting a vital aspect of the synchronization process.  
The synchronization via substitution method in the Colpitts chaos oscillator can be established by utilizing voltage across capacitors \(C_{1}\) or \(C_{2}\). The drive oscillator generates three state variables (\(V_{C1}\), \(V_{C2}\), and \(I_{L}\)). After the synchronization, the response oscillator generates only two state variables. If \(V_{C2}\) is used for synchronization, the response chaos oscillator generates \(V^{\prime}{}_{C1}\) and \(I^{\prime}{}_{L1}\). In other solutions, the response chaos oscillator generates \(V^{\prime}{}_{C2}\) and \(I^{\prime}{}_{L}\).  
An example of the synchronization in the chaos oscillator is depicted in Figure 3. The signal \(V_{C1}\) from the drive oscillator replaces the voltage across capacitor \(C^{\prime}{}_{1}\) in the response oscillator. Initially, for the first two milliseconds, both the drive and response chaos oscillators operate independently. However, after 2 ms, the drive-response synchronization process is complete, enabling the response oscillator to replicate the behavior of the drive oscillator.  
The synchronization between the drive and response circuits of chaotic oscillators must be evaluated. Pearson's correlation coefficient \(\beta\) was chosen for this purpose since it can evaluate how similar the signals are on a scale of \(-1\) to \(1\). The Pearson's correlation coefficient \(\beta\) is calculated as follows:  
\[\beta=\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}-\overline {y}\right)}{\sqrt{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}\sqrt{ \sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}}} \tag{4}\]  
where \(x\) represents the drive circuit signal and \(y\) represents the identical element signal in the response circuit. When signals match identically, \(\beta\) equals \(1\). In the case of identical signals but with opposite phases, \(\beta\) takes a value of \(-1\). For signals that differ and are dissimilar, \(\beta\) equals \(0\).  
The correlation coefficient between the voltages across their respective components converges toward one when the circuits achieve synchronization. This significant correlation indicates that the drive-response system's two chaotic oscillators are well-synchronized.  
Figure 3: Colpitts Chaos oscillator drive-response synchronization, where \(V_{C1}\), \(V_{C2}\), and \(I_{L}\) represent signals in drive circuit and \(V^{\prime}{}_{C1}\), \(V^{\prime}{}_{C2}\), and \(I^{\prime}{}_{L}\) represent signals in response circuit.  
An operational amplifier working in voltage-follower mode establishes the implementation of chaotic synchronization, enabling data transmission.