## 2 Working Models  
### Colpitts Chaotic Synchronization  
The Colpitts circuit is a classic harmonic oscillator, being one of the most widely used electronic designs for the development of harmonic data transmission for almost a century. Interestingly, in 1994 it was reported that a parametric set of circuit values creates chaotic dynamics [27]. The Colpitts circuit consists of a single bipolar junction transistor (BJT), which is biased in its active region [27; 28]. A combination of inductor \(L\), capacitance divider, and resistance \(R\) was used as a feedback network. Figure 1 shows a circuit schematic. Figure 1: Colpitts circuit. Based on the respective Kirchhoff analysis, the dynamics of the Colpitts oscillators with symmetrical feedback operated in the forward active and cut-off regions can be described by a system of three autonomous state equations given by:  
\[\left\{ \begin{array}{c}C_{1}\frac{dV_{CE}\ (t)}{dt}=I_{L}(t)-I_{C}(t)\\ C_{2}\frac{dV_{BE}(t)}{dt}=-\frac{V_{EE}+V_{BE}(t)}{R_{EE}}-I_{L}(t)-I_{B}(t)\\ L\frac{dI_{L}(t)}{dt}=V_{CC}-V_{CE}(t)+V_{BE}(t)-I_{L}(t)R_{L}\end{array} \right. \tag{1}\]  
where \(V_{CC}\) and \(V_{EE}\) are the positive and negative power supply voltages, respectively. \(V_{CE}(t)\) and \(V_{BE}(t)\) are the collector-emitter voltage and the base-emitter voltage correspondingly and \(I_{L}(t)\) is the current flowing through \(L\). The Kirchhoff analysis also shows that both the base and collector currents \(I_{B}(t)\) and \(I_{C}(t)\), both depend on \(V_{BE}(t)\). Let \(V_{TH}\) denote the transistor's voltage threshold and \(R_{ON}\) its corresponding internal resistance in the conducting mode, the BJT's switching between the cut-off and active modes result in the jump of the Colpitts dynamics. Finally, the term \(\beta\) is the forward current gain of the transistor. The switching law that represents the transistor behavior can be modelled as a two-segment piecewise linear function:  
\[I_{B}(t)=\left\{ \begin{array}{cc}0&if\ V_{BE}(t)\leq V_{TH}\\ \frac{V_{BE}(t)-V_{TH}}{R_{ON}}\ if\ V_{BE}(t)>V_{TH}\end{array} \right.,I_{C}(t )=\beta I_{B}(t), \tag{2}\]  
In this study, we have adopted the circuit parameters described in [28], obtained from experimental setups. The parameters are given in Table 1. The single bipolar junction transistor employed is the model 2 SC1740 with a cutoff frequency of 180 MHz, from which the operating values are extracted. The interested reader can refer to [28] for further detail. Figure 2 shows the voltage and current waveforms for all the states given in (1) when operating in chaotic mode, that is, when using the parameters given in Table 1. \(V_{BE}(t)\) is highlighted since it represents the transmitted carrier signal, which is also used at the Rx to ensure coupling between both chaotic trajectories. Further detail is given in the upcoming paragraphs. As follows we show Colpitts chaotic attractor projection, obtained by plotting the states against each other, shown in Figure 3. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Values** \\ \hline \(C_{1}\) (nF) & 58 \\ \(C_{2}\) (nF) & 58 \\ \(L\) (\(\mu\)H) & 100 \\ \(R_{L}\) (\(\Omega\)) & 36 \\ \(R_{EE}\) (\(\Omega\)) & 430 \\ \(V_{CC}\) (V) & 5 \\ \(V_{EE}\) (V) & \(-5\) \\ \(R_{ON}\) (\(\Omega\)) & 375 \\ \(\beta\) & 230 \\ \(V_{TH}\) & 0.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Colpitts circuit parameters. As an illustrative example, we show the chaotic synchronization method employed for an ideal channel. We assume that, both Tx and Rx have identical parameters except for the initial conditions, which are set at different values. Figure 3: Colpitts attractor projections. Figure 2: Colpitts chaotic time series. For this we use the identical (or complete) chaotic synchronization methodology proposed in [29], where VBE_TX(t) (i.e., the chaotic carrier signal) replaces VBE_RX(t) and directly dependent terms of IB_RX(t) and IC_RX(t) in the 1 st and 3 rd equations of (3). Note that, VBE_RX(t) is left unchanged in the 2 nd equation.