## 3 Experimental Evidence of Synchronization between Rossler Systems
### Adaptive Generalized Synchronization between Circuit and Computer Model
#### 3.3.2 Circuit Operation  
To validate the correct operation of the circuit, its phase portrait and bifurcation diagram were obtained. For the phase portrait of the analog circuit (Figure 6) with parameters \(a=0.2\), \(b=0.2\), \(c=5.7\), refer to Figure 7a, which represents the recognizable chaotic scroll. In the bifurcation diagram (Figure 7b) which demonstrates chaotic attractor points as a function of parameter \(c\), Feigenbaum sequences of bifurcations characterizing the Rossler system are also recognizable. Data acquisition and processing was performed using the NI ELVIS III station and the software developed in the LabVIEW 2019 environment. However, the specific values of the nonlinearity parameters at which different oscillation modes are observed in the analog circuit differ from those in the numerical simulation. To ensure consistency between the diagrams for parameter \(c\), the value of parameter \(b\) in the analog circuit has been adjusted. A feature of the Rossler system that makes it difficult to implement in the analog computer is the large ratio between the values of the variable \(z\) amplitude and parameter \(b\). Therefore, in our implementation, the value \(b/40=0.005\) V was applied to the summing input of U\({}_{5}\) multiplier. This value is presumably of the same order of magnitude as the weak output bias present in the multiplier and also is strongly distorted by analog noise. Thus, when specifying the value \(b\), this offset must be taken into account. In our case, the bias had a value of about 0.0025 V. Any adjustments of the value of parameter \(b\) do not lead to a complete correspondence between bifurcation diagrams, since there are other nonlinearities and errors in specifying the values of components in the analog circuit that affect its operation; in particular, the amplitudes of the attractors differ. Thus, system (19), which is a linear transformation of system (10), does not fully characterize the dynamics of the circuit. The exact identification of non-linear components is challenging. Therefore, complete synchronization between the circuit and the mathematical model is not possible. The same result was achieved in one of our previous works [22], where the Rossler system was also considered and error attractors in case of Pecora-Carroll synchronization [1] were obtained. Note that in [22], a slightly modified Rossler oscillator with an additional parameter \(d\) was studied; additionally, another circuit was designed with the parameter \(c\) defined as the ratio between two resistors, not as an external voltage. 3.3 Adaptive Generalized Synchronization between Circuit and Numerical Model with Respect to Numerical Method  
The experimental study of adaptive generalized synchronization between the circuit and the computer model was conducted using the following parameters of the slave system: \(x_{s,1}=2\), \(y_{s,1}=-4\), \(z_{s,1}=-2\), initial parameters \(\hat{a}_{1}=0.2\), \(\hat{b}_{1}=0.6\), \(\hat{c}_{1}=5.25\). [PERSON]'s integration method was used, and the step-size was \(h=0.0464\). The auxiliary slave system was similar. Figure 8 presents the time series of the error between the circuit and computer model, taken as the \(L_{2}\)-norm of the error in dynamics, and the time series of estimated parameters. One can clearly see that the estimated parameters are oscillating and do not converge to steady-state values. This behavior shows that the perturbations caused by the introduced errors are significant and cannot be tracked by the controller well. However, the overall stability of the solution is not violated. Figure 7: **(a)** Attractor produced by the circuit. **(b)** Bifurcation diagrams of overlaid numerical (Runge–Kutta 4) and analog systems. Parameter \(b=0.2\) was used for the numerical system, and parameter \(b=0.3\) for the analog system. Figure 9 presents the synchronization error between master and slave systems for different integration methods, smoothed with the median filter. There is a clear dependence of the averaged error on the integration method.