# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 3 Example Illustration and Demonstrated Results
### Simulation Results and Discussion  
Figure 2 shows that the trajectories in the two-dimensional x1, x3-spaceofthe drive system are in the red line; the response system as the tracing system is in blue. The drive system with initial condition, \(\mathbf{x_{m0}}=(0.02,0.01,\ 0.03)\), differs from the response system, \(\mathbf{x_{s0}}=(5,\ 10,\ 10)\). The difference between the drive and response systems was larger than in other articles in which the initial condition alteration of a drive-response system was closed or not mentioned as in [12; 13; 14; 15; 16; 17; 18]. The trajectory of the response systems quickly approached that of the drive system after their initial condition and the approximation nearby the drive system, just as the iterative learning-controlled law containing the information from the drive system dominated the response system along the trajectory of the drive system. According to Equation (2) and Lemma 2, the convergent error dynamics \(\dot{\mathbf{\Delta}}^{(\mathbf{k})}\) should be less than or convergent to zero, and the behavior is demonstrated in Figure 3. The components in \(\dot{\mathbf{\Delta}}^{(\mathbf{k})}\) were convergent to zero with averages of [\(-0.01341\), \(-0.0122\), and \(-0.1796\)] for each state of the system,  
Figure 2: The trajectories in two-dimensional x1, x3-space of systems. respectively, satisfying the error convergent criterion. Three state vectors of error dynamics were vibration in the bounded interval as well, to verify the bounded error of each iteration and to design an appropriate controller by the ILC rule in Equation (2) and Lemma 2. The behaviors of the ILC rule in Figure 4 show that the three components were always non-increasing and approached zero, and exhibited that the components were zero-central vibration in a bounded interval with average [-0.00919, 0.00171, -0.00469]. According to Equation (6), when the tracking error is an approximation to zero, the ILC law should approach zero in Lemma 2 and be a non-increasing and bounded function in Corollary 1. The simulation results of Equation (6) have already examined Lemma 2 and Corollary 1. The behaviors were not always identical to different ILC rules as shown in Figure 5. It is an important issue and challenges to select an eligible ILC rule. In Figure 5a, the \(\mathbf{B_{1}}\) in Equation (6) was chosen as the identity matrix, and \(\mathbf{B_{1}}\) in Figure 5b as a diagonal matrix [0.1, 0.1, 0.92]. In both Figure 5a,b, one of the components rose sharply and continued to vibrate in a bounded interval. Although the other components gradually decreased in the previous ILC rule, the ILC rule still makes the response system under an unpredictable situation. The behavior of the derivative of the Lyapunov function is demonstrated in Figure 6 and corroborates both Theory 1 and Lemma 2 by Equations (11) and (12). The data in Table 1 is the derivative of the Lyapunov function to count the number in which the value of the Lyapunov function in this article was zero or negative. The number in Table 1 was the closed relative to the different parameters \(\mu\) in Equation (10). From the average in the table, most of the values of the Lyapunov function tended to be negative. Hence, the average of values of the Lyapunov function became positive when \(\mu<0\). The most significant phenomenon was to regard the ILC rule to find an appropriate linear combination  
Next, Figure 8 shows the synchronization error for each component of the systems in Equations (15) and (16). The synchronization error of each component indicated that it was convergent to zero and vibrated in the neighborhood of zero. The average of the synchronization errors in 12,000 time steps were (-0.00049, -0.00077, and -0.00199) for each component. The exhibition of vibration in Figure 8 matched all components in Figure 3, in which the error dynamics should be convergent to zero when the tracking error approaches zero.