## 3 Experimental Evidence of Synchronization between Rossler Systems
### Tracking an Additional Parameter in Adaptive Generalized Synchronization  
Modify the Rossler system (10) as follows:  
\[\begin{cases}\dot{x}&=-\mu y-z,\\ \dot{y}&=x+ay,\\ \dot{z}&=b+z(x-c),\end{cases} \tag{20}\]  
where \(\mu\) is the novel parameter, which is not tracked by the controller (13). This parameter is easily tuned in the circuit by changing the only resistor \(R_{1}\). In our experiment, the following values of \(R_{1}\) were used: 5 kOhms, 8 kOhms, 9 kOhms, 10 kOhms, 10.5 kOhms, 11 kOhms, 12.8 kOhms and 20 kOhms. Different attractors of the Rossler system corresponding to these values are given in Figure 12.  
Parameter \(\mu\) values differing from 10 kOhms introduce additional error in the implemented analog Rossler system. Attractors \(\mathcal{A}_{c}\subset\mathbb{R}^{3}\) of the error in dynamics \(\mathbf{e}(n)\) between the circuit implementation of the master system (20) and computer model of the slave system (12) are given in blue in Figure 13.  
Then, we equip the slave system with an additional adjustable parameter \(\hat{\mu}\), which now reads as  
\[\begin{cases}\dot{x}_{s}&=-\hat{\mu}y_{s}-z_{s}-u_{1},\\ \dot{y}_{s}&=x_{s}+\hat{\mu}y_{s}-u_{2},\\ \dot{z}_{s}&=\hat{b}+z_{s}(x_{s}-\hat{c})-u_{3}.\end{cases} \tag{21}\]  
Modify the adaptive controller in the following way:  
\[\begin{cases}u_{1}&=e_{1}+(z_{m}-1)e_{3}+(1-\hat{\mu})e_{2},\\ u_{2}&=(1+\hat{\mu})e_{2},\\ u_{3}&=(1+x_{s}-\hat{c})e_{3},\end{cases} \tag{22}\]  
Figure 12: Attractors acquired from the Rössler modeling circuit with mathematical parameter values \(a=b=0.2\), \(c=5.7\) and the altered resistor \(R_{1}\). Here, \(\mu=R_{0}/R_{1}\). The case \(\mu=1\) corresponds to a normal Rössler oscillator; other cases show the deliberate corruption of the system.  
with the following update rule for parameters:  
\[\begin{cases}\dot{a}&=-y_{m}e_{2},\\ \dot{b}&=-e_{3},\\ \dot{c}&=z_{m}e_{3},\\ \dot{\mu}&=y_{m}e_{1}.\end{cases} \tag{23}\]  
After substituting (20)-(23) into Equation (9) from [27], it is easy to see that the slave system (21) completely synchronizes with the system (20).  
The actual error in synchronization between the circuit and digital implementation of the system (20) using [PERSON]'s method with a step-size of \(h=0.0518\) s is given in red in Figure 12. This is substantially smaller in comparison with the error of the system (21). To obtain a quantitative measure of error attractors, we used the attractor \(L_{1}\)-volume \(V_{L_{1}}(\mathcal{A}_{e})=\sum_{j=x,y,z}(\max_{n}(e_{j}(n))-\min_{n}(e_{j}(n )))\). Its estimations are given in Figure 14.  
One can see that additional system perturbation with the parameter \(\mu\) increases the synchronization error when this parameter is not tracked, and the new value of \(\mu\) sufficiently differs from 1. At the same time, tuning the corresponding parameter \(\hat{\mu}\) allows a notable reduction of the synchronization error.  
To estimate the error in the parameter \(\mu\) identification, the median value over 9000 samples of \(\hat{\mu}(n)\) was computed. The estimated \(R_{1}\) was calculated as  
\[\hat{R}_{1}=\frac{R_{0}}{M_{n=1.9\cdot 10^{3}}(\hat{\mu}(n))},\]  
where \(M\) is the median and \(R_{0}\) is 10 kOhms. Obtained results are given in Figure 15.  
Figure 13: Error attractors between the circuit and the computer model: blue for the slave system (12) and red for the slave system (21).  
Although there was no complete synchronization between analog and computer systems, the relative error of parameter identification was on average less than 1.5%. This shows the practical applicability of this approach as a parameter identification technique.