## 3 Experimental Evidence of Synchronization between Rossler Systems
### Adaptive Generalized Synchronization between Circuit and Computer Model  
#### 3.3.1 Circuit Implementation  
The implementation of the chaotic circuit based on the Rossler system is shown in Figure 6. This circuit was designed with the classical principles of analog computer design. The circuit consists of three integrators made using the operational amplifiers U\({}_{1}\), U\({}_{2}\) and U\({}_{4}\), each corresponding to the line of the differential Equation (10). Figure 4: Synchronization error between computer models, smoothed with a median filter; the window size is 600 samples. Figure 5: Synchronization error between slave systems for different integration methods, smoothed with a median filter; the window size is 600 samples. We start circuit synthesis with selection of \(R_{0}\) and \(C_{0}\), defining the simulation time constant \(\tau=R_{0}C_{0}\). In our case, \(R_{0}=10\) k\(\Omega\) and \(C_{0}=10\) nF. Each state variable or constant term in the original equation is connected via a resistor to an integrator. If a state variable is multiplied by some constant \(\xi\), then the corresponding resistor value decreases by \(\xi\) times; i.e., \(R=R_{0}/\xi\). If \(\xi=1\), the resistor \(R_{0}\) is used. This principle allows the easy construction of circuit parts corresponding to the first and second lines of (10). When considering the last line, one should keep in mind that the amplitudes of state variables require a decrease of \(M\) times compared to the original Rossler oscillator state variables. This is due to the fact that tge \(z\) variable in the Rossler system with standard parameters can exceed not only 10 but even 20, which in analog implementation means 20 V, while an allowable voltage range is commonly from \(-10\) V to \(+10\) V. Since the similar scale constant \(M\) is used for all state variables, only the last line of (10) needs to be changed. Let us write it as follows:  
\[M\hat{\underline{z}}=V_{b}+M\hat{\underline{z}}(M\hat{\underline{x}}-V_{c}), \tag{15}\]  
where \(x=M\hat{\underline{x}}\) and \(y=M\hat{\underline{y}}\) are the state variables with no scaling, and \(V_{b}\) and \(V_{c}\) are voltages with values that are mathematically equal to tge values of appropriate parameters in numerical system: \(V_{b}=b\), \(V_{c}=c\). Then,  
\[\hat{\underline{z}}=V_{b}/M+\hat{\underline{z}}(M\hat{\underline{x}}-V_{c}). \tag{16}\]  
Recall also that multiplier AD633 (Analog Devices, Norwood, MA, US), the element U\({}_{5}\) in Figure 6, acts as follows:  
\[W=(X_{1}-X_{2})(Y_{1}-Y_{2})/10+Z, \tag{17}\]  
i.e., the output voltage \(W\) is the sum of the input voltage \(Z\) and the product of the input voltages \((X_{1}-X_{2})\) and \((Y_{1}-Y_{2})\) divided by 10. This scaling is necessary so that the  
Figure 6: Circuit implementing the Rössler chaotic system with parameters \(b\) and \(c\) defined as input voltages. Circuit state variables \(X\), \(Y\) and \(Z\) presented as voltages are down-scaled by \(M=4\) compared to their numerical counterparts. multiplication operation does not lead to the appearance of impossible voltages in the circuit. Taking this into account, we rewrite the third line of (10):  
\[\dot{z}=10\left(\frac{V_{b}}{10M}+\underbrace{\frac{\dot{z}}{10}(M\hat{x}-V_{c})} _{\hat{\psi}}\right), \tag{18}\]  
where the term \(\psi\) is the direct output from the U\({}_{5}\) (AD633) multiplier. From (18), it follows that \(V_{b}\) requires division by \(10M\) before application to the \(Z\) input of the multiplier. Furthermore, the factor 10 at the beginning of the line defines the resistor \(R_{13}=R_{0}/10\) between the multiplier U\({}_{5}\) and the integrator U\({}_{4}\). The exact equation of the whole circuit in Figure 6 is  
\[ \begin{cases}C_{1}\dot{\hat{x}}=-\frac{\hat{y}}{R_{1}}-\frac{\dot{z}}{R_{2}},\\ C_{2}\dot{\hat{y}}=\frac{\hat{x}}{R_{3}}+\frac{\hat{y}}{R_{4}},\\ C_{3}\dot{\hat{z}}=\frac{R_{0}V_{b}}{R_{0}R_{13}}+\frac{\dot{z}}{10R_{13}}\left( \frac{R_{12}\hat{x}}{R_{10}}-\frac{R_{12}V_{c}}{R_{11}}\right).\end{cases}