# Synchronization of Analog-Discrete Chaotic Systems for Wireless Sensor Network Design
## 2 Chaos Oscillators, Discrete Models and Chaotic Synchronization
### Vilnius Chaos Oscillator  
The second chaos oscillator for this work is \"A simple chaotic oscillator for educational purposes\" [26], which is referred to as \"Vilnius chaos oscillator\". The circuit diagram, LTspice model, and fabricated prototype of this oscillator are shown in Figure 6.  
The dynamics of the oscillator are described with a system of differential equations in (6). The state variables are the voltages across the capacitors \(C_{1}\) and \(C_{2}\) as well as the current through the inductor \(L_{1}\).  
\[\left\{\begin{array}{l}C_{1}\frac{d\upsilon_{C_{1}}}{dt}=i_{L_{1}},\\ L_{1}\frac{d\upsilon_{L_{1}}}{dt}=(k-1)R_{1}i_{L_{1}}-\upsilon_{C_{1}}- \upsilon_{C_{2}},\\ C_{2}\frac{d\upsilon_{C_{2}}}{dt}=i_{0}+i_{L_{1}}-i_{D},\end{array}\right. \tag{6}\]  
where \(i_{D}\) is the current of the diode \(D_{1}\); \(i_{0}\) is the current of the resistor \(R_{4}\); \(k\) is the gain of \(O\)\(A\). The nominals and parameters of the other elements are: \(C_{1}=1\) nF; \(C_{2}=150\) pF; \(L_{1}=1\) mH; \(R_{1}=1\) k\(\Omega\); \(R_{2}=10\) k\(\Omega\); \(R_{3}=6\) k\(\Omega\); \(R_{4}=20\) k\(\Omega\); \(V_{1}=5\) V; \(k=\frac{R_{2}}{R_{3}^{2}}\). The nonlinearity of the system is introduced with the diode current's equation in (7).  
\[i_{D}=i_{s}\cdot(\exp(\frac{eV_{C_{2}}}{k_{B}T})-1), \tag{7}\]  
Like in the previous case, the dimensionless variables in (9) are introduced, resulting in (8). Next, the differentiation is taken over \(\theta\)--time \(t\) scaled by the time constant \(\tau\) (10) and (11).  
\[\left\{\begin{array}{l}\frac{dx}{dt}=y,\\ \frac{dy}{dt}=a\cdot y-x-z,\\ e\cdot\frac{dz}{dt}=b+y-c(\exp(z)-1),\end{array}\right. \tag{8}\]  
Figure 5: RC chaos oscillator’s hardware prototype’s attractor \(x\)-\(y\) projection (**a**), \(y\)-\(z\) projection (**b**) and \(x\)-\(z\) projection (**c**).  
Figure 6: Circuit diagram of RC chaos oscillator (**a**), the fabricated prototype (**b**), and LTspice model (**c**).  
\[\begin{array}{l}x=\frac{v_{C_{1}}}{V_{T}},\qquad y=\frac{\rho\cdot i_{1}}{V_{T}}, \quad z=\frac{v_{C_{2}}}{V_{T}},\\ V_{T}=\frac{k_{B}T}{\epsilon},\qquad\rho=\sqrt{\frac{L_{1}}{C_{1}}},\quad \epsilon=\frac{C_{2}}{C_{1}},\\ a=(k-1)\frac{R_{1}}{\rho},\quad b=\frac{v_{B}}{V_{T}},\quad c=\frac{c_{B}}{V_{ T}},\\ \left\{\begin{array}{l}\frac{dx}{d\theta}=y,\\ \frac{dy}{d\theta}=a\cdot y-x,\\ \epsilon\cdot\frac{dz}{d\theta}=b+y-c(\exp(z)-1),\end{array}\right.\\ \theta=\frac{t}{\tau},\quad\tau=\sqrt{L_{1}\cdot C_{1}}\end{array} \tag{9}\]  
Finally, the Euler-Cromer numerical integration [35] is taken, acquiring the approximate discrete solution of the system in (12).  
\[\left\{\begin{array}{l}x_{n+1}=a\cdot y_{n}\cdot\Delta\theta+x_{n},\\ y_{n+1}=y_{n}\cdot(1+a\cdot\Delta\theta)-x_{n}\cdot\Delta\theta-z_{n}\cdot \Delta\theta,\\ z_{n+1}=\frac{\Delta\theta}{\epsilon}(b+y_{n}-c\cdot(\exp(z_{n})-1))+z_{n}, \end{array}\right. \tag{12}\]  
Like the RC oscillator, the following circuit was implemented in hardware shown in Figure 6b. The discrete model based on system (11) was implemented in MATLAB, as well as LTspice (Figure 6c). Figures 7-9 demonstrate the attractor projections of the circuit acquired from the abovementioned models. The state variable signals acquired from LTspice and recorded from hardware are resampled and scaled to match the discrete model like with the RC chaos oscillator. The figures demonstrate that the attractors are similar across the models, with the differences being in the amplitudes and offsets of the signals.