## 2 Chaos Oscillators
### Chaotic Synchronization  
Chaotic synchronization is fundamental for utilizing chaotic oscillators in secure communication systems [47]. It enables the secure transmission of information by ensuring that only synchronized systems can accurately extract data from received signals. Among the various synchronization methods, the substitution method, also known as Pecora-Carroll synchronization [48], is particularly noteworthy for its simplicity and effectiveness in both analog and discrete chaotic systems. The substitution method replaces one of the state variables in the response chaotic oscillator with its corresponding variable from the drive oscillator. In this approach, the response oscillator is driven by a signal from the drive circuit, aiming to align their chaotic trajectories. Pearson's correlation coefficient is often employed to evaluate the quality of synchronization. This metric quantifies the similarity between the signals from the drive and response oscillators. The coefficient, denoted as \(\beta\), is calculated using the formula:\[\beta=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_{ i}-\bar{x})^{2}\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}}, \tag{6}\]  
where \(x\) is the signal from the drive oscillator and \(y\) is the signal from the response circuit. If the signals are identical, \(\beta\) equals 1; if they are inverted, \(\beta\) is \(-1\); and if \(\beta\) is 0, the signals are uncorrelated, indicating no similarity. The same node is selected for synchronization in both the drive and response chaos oscillators to implement the substitution method. An operational amplifier configured as a voltage repeater (buffer circuit) is commonly used for analog chaos oscillator synchronization [49; 50; 51]. This configuration determines the direction of synchronization and allows the signal in the response circuit to be substituted with the corresponding signal from the drive oscillator. Since chaotic oscillators typically have multiple nodes, synchronization can be achieved in various configurations by selecting different nodes for substitution. An example illustrating unsynchronized and synchronized signals is presented in Figure 3. In Figure 3a, the XY plot shows the unsynchronized signals from the same node of identical chaos oscillators. The lack of order is noticeable, as the data points are scattered randomly. In contrast, Figure 3b presents an XY plot of a synchronized drive-response system. The x-axis represents the signal from the drive chaos oscillator, while the y-axis shows the signal from the same node in the response oscillator. The data points align along a diagonal line, demonstrating that the voltage in the response oscillator corresponds precisely to the voltage in the drive oscillator. Such a result is indicative of a successfully synchronized system. In the Colpitts chaos oscillator (Figure 1), synchronization can be established using two configurations, i.e., \(v_{C1}\) or \(v_{C2}\), as synchronization signals [52]. For instance, if \(v_{C1}\) is chosen for synchronization, the drive oscillator generates all three state variables (\(v_{C1}\), \(v_{C2}\), and \(i_{L1}\)), while the response oscillator generates \(v_{C2}\) and \(i_{L1}\), with \(v_{C1}\) being substituted by the corresponding signal from the drive oscillator. In the Vilnius chaos oscillator (Figure 2), synchronization can be achieved using \(v_{C1}\), \(v_{C2}\), or \(v_{R1}\) (representing the current \(i_{L}\)) as a synchronization signal. These multiple synchronization points offer flexibility for future applications. As synchronization in chaotic oscillators forms the basis for their integration into communication systems, it is essential to assess the noise resilience of synchronization  
Figure 3: XY plot of drive-response system (**a**) without synchronization and (**b**) with chaotic synchronization applied. across various configurations. Understanding how synchronization performance varies in noisy environments is critical for ensuring reliable communication.