## 3 Experimental Evidence of Synchronization between Rossler Systems  
### Adaptive Complete Synchronization between Numerical Models  
As the sample model, we consider the Rossler chaotic system that is described by the following system of ordinary differential equations (ODE):  
\[\begin{cases}\dot{x}&=-y-z,\\ \dot{y}&=x+ay,\\ \dot{z}&=b+z(x-c),\end{cases} \tag{10}\]  
where \(x,y,z\) are state variables and \(a,b,c\) are nonlinearity parameters.  
In [27], [PERSON] proposed a technique to identify three unknown parameters of a system (10) using adaptive control. According to such a method, the master (\(m\)) and slave (\(s\)) systems are defined as  
\[\begin{cases}\dot{x}_{m}&=-y_{m}-z_{m},\\ \dot{y}_{m}&=x_{m}+ay_{m},\\ \dot{z}_{m}&=b+z_{m}(x_{m}-c),\end{cases} \tag{11}\]  
and  
\[\begin{cases}\dot{x}_{s}&=-y_{s}-z_{s}-u_{1},\\ \dot{y}_{s}&=x_{s}+\hat{a}y_{s}-u_{2},\\ \dot{z}_{s}&=\hat{b}+z_{s}(x_{s}-\hat{c})-u_{3},\end{cases} \tag{12}\]  
where \(u_{1}\), \(u_{2}\) and \(u_{3}\) are control laws and \(\hat{a}\), \(\hat{b}\) and \(\hat{c}\) are estimated parameters. Based on the [PERSON] stability theory, synchronization of the master and slave systems is achieved under the control laws given by  
\[\begin{cases}u_{1}&=e_{1}+(z_{m}-1)e_{3},\\ u_{2}&=(1+\hat{a})e_{2},\\ u_{3}&=(1+x_{s}-\hat{c})e_{3},\end{cases} \tag{13}\]with the update rule for parameters described as follows:  
\[\begin{cases}\dot{a}&=-y_{m}e_{2},\\ \dot{b}&=-e_{3},\\ \dot{c}&=z_{m}e_{3},\end{cases} \tag{14}\]  
where \(\mathbf{e}=(e_{1},e_{2},e_{3})^{\top}\) is the error vector between (11) and (12):  
\[\begin{cases}e_{1}(t)&=x_{s}(t)-x_{m}(t),\\ e_{2}(t)&=y_{s}(t)-y_{m}(t),\\ e_{3}(t)&=z_{s}(t)-z_{m}(t).\end{cases}\]  
To verify the considered approach for parameter estimation, we simulated the master and slave systems using the Dormand-Prince eighth-order integration method with a fixed integration step of \(h=0.01\) within 500 seconds. For systems (11) and (12), the initial conditions were chosen as \(x_{m}=-1\), \(y_{m}=0\), \(z_{m}=1\) and \(x_{s}=2\), \(y_{s}=-4\), \(z_{s}=-2\) while the parameters were \(a=0.2\), \(b=0.6\) and \(c=5.25\). The initial values of three parameters for the slave system were equal \(\dot{a}=\dot{b}=\dot{c}=1\). Figure 2 presents the results of adaptive synchronization between two computer models simulated with the eighth-order integration method. Parameters are fitted exactly, and the error reaches the machine epsilon in double precision.  
The numerical simulation also shows that the lowest reachable error in a complete adaptive synchronization depends on the accuracy of the integration method. Figure 3 presents the error \(\mathbf{e}(t)\) dynamics between a master system implemented with an eighth-order Dormand-Prince method and a slave system implemented with various methods of orders 1 and 2 obtained in a numerical experiment. We considered eight integration methods listed below with the corresponding notations: the Euler method (_Euler_), Euler-Cromer method (_Euler_-_Cromer_), semi-implicit CD-method (_CD_), [PERSON]'s method (_Heun_), implicit midpoint method (_IMP_), explicit midpoint method (_EMP_) and trapezoidal rule (_Trapezoidal_).  
The best results are achieved using second-order Heun and trapezoidal methods.  
Figure 2: (**a**) Synchronization error between master and slave system. (**b**) Evaluation of parameters for the slave system.