# Synchronization of Analog-Discrete Chaotic Systems for Wireless Sensor Network Design
## 2 Chaos Oscillators, Discrete Models and Chaotic Synchronization
### RC Chaos Oscillators  
The first chaos oscillator for this work is the \"RC chaos oscillator\" [30]. The circuit diagram, LTspice model, and fabricated prototype of this oscillator are shown in Figure 2.  
The dynamics of the circuit are described with a system of differential equations in (1). The state variables are voltages across the capacitors \(C_{1}\), \(C_{2}\), and \(C_{3}\). \[\left\{ \begin{array}{l}\frac{dx}{dt}=-x+(k_{1}-1)\cdot y-(k_{1}-1)\cdot z,\\ \frac{dy}{dt}=-x+(k_{1}-2)\cdot y-(k_{1}-1)\cdot z,\\ \frac{dz}{dt}=yk_{1}+(k_{1}+\alpha)\cdot z+\beta\cdot(z-1)\cdot H(z-1).\end{array} \right. \tag{1}\]  
\[x=\frac{v_{C_{1}}}{v*},\ \ y=\frac{v_{C_{2}}}{v*},\ \ z=\frac{v_{C_{3}}}{v*},\ \ \ v*= \frac{v_{0}}{(k_{2}-1)},\ \ \ \alpha=\frac{R}{R_{6}},\ \ \ \beta=\frac{R}{R_{8}},\ \ \ k_{1}=\frac{R_{3}}{R_{4}}+1,\ \ \ k_{2}=\frac{R_{7}}{R_{8}}+1, \tag{2}\]  
where \(v_{C_{1}}\), \(v_{C_{2}}\), and \(v_{C_{3}}\) are voltages across the capacitors \(C_{1}\), \(C_{2}\), and \(C_{3}\), respectively; \(v_{0}\)--voltage across the diode \(D_{1}\); \(k_{1}\)--gain of the operational amplifier \(OA_{1}\); \(k_{2}\)--gain of the operational amplifier \(OA_{2}\); \(H()\)--Heaviside function. The elements have the following values: \(R=R_{1}=R_{2}=11\) k\(\Omega\); \(R_{3}=9.1\) k\(\Omega\); \(R_{4}=2\) k\(\Omega\); \(R_{5}=R_{7}=2.7\) k\(\Omega\); \(R_{6}=1.1\) k\(\Omega\); \(R_{8}=780\)\(\Omega\); \(C=C_{1}=C_{2}=C_{3}=1.3\) nF. The implemented prototype in Figure 2b uses the 1N4148 diode and TL082 operational amplifiers. These are general off-the shelf components, as the circuit has no specific criteria for these elements. The system (1) describes the dynamics of the oscillator in continuous time (the analog circuit). In this case, the fundamental frequency of oscillations is defined by the time constant \(\tau(4)\). The discrete model of the oscillator would represent the digital circuit that exhibits similar behavior as its analog counterpart. To derive a discrete model of the oscillator, several modifications to (1) must be made. First, the differentiation is taken over \(\theta\), which is the time \(t\) scaled by the time constant \(\tau\). \[\left\{ \begin{array}{l}\frac{dx}{d\theta}=-x+(k_{1}-1)\cdot y-(k_{1}-1) \cdot z,\\ \frac{dy}{dt}=-x+(k_{1}-2)\cdot y-(k_{1}-1)\cdot z,\\ \frac{dz}{d\theta}=yk_{1}+(k_{1}+\alpha)\cdot z+\beta\cdot(z-1)\cdot H(z-1). \end{array} \right. \tag{3}\]  
\[\theta=\frac{t}{\tau},\ \ \ \tau=R\cdot C \tag{4}\]  
Figure 2: Circuit diagram of RC chaos oscillator (**a**), the fabricated prototype (**b**), and the LTspice model (**c**). The second step applies the Euler-Cromer numerical integration [35], acquiring the approximate discrete solution of the system (5). \[\left\{ \begin{array}{l}x_{n+1}=(1-\Delta\theta)\cdot x_{n}+(k_{1}-1)\cdot y_{n} \cdot\Delta\theta-(k_{1}-1)\cdot z_{n}\cdot\Delta\theta,\\ y_{n+1}=-x_{n}\cdot\Delta\theta+((k_{1}-2)\cdot\Delta\theta+1)\cdot y_{n}-(k_ {1}-1)\cdot z_{n}\cdot\Delta\theta,\\ z_{n+1}=y_{n}k_{1}\cdot\Delta\theta+((k_{1}+\alpha)\cdot\Delta\theta+1)\cdot z _{n}+\beta\cdot(z_{n}-1)\cdot H(z_{n}-1)\cdot\Delta\theta,\end{array} \right. \tag{5}\]  
where \(\Delta\theta\) is the time step and \(n\) is the index of the discrete solution. The system (5) represents the circuit's dynamics in discrete form. In this case, the frequency of chaotic oscillations is not defined by \(\tau\) but by the discrete circuit's clock frequency. The \(\Delta\theta\) does not represent real time in this case and is dimensionless but must be selected carefully to ensure chaotic behavior. It is important to note that this research works with the discrete model itself, and not its full implementation as a digital circuit. For this reason, the discrete model based on system (5) was implemented in MATLAB. The fabricated prototype of the circuit is shown in Figure 2b. The variable resistor is used to set the 780 \(\Omega\) resistance. This work also uses the LTspice model of the circuit (Figure 2c).