PDF《Rssler systems with uncertain》

5 2 ? a2

4 ? b2

4 ? k1 )

0 0

0 (?a ?

2 ? k2)

0 0

0 ( c ?

5 2 ) ? ? ? ? ? ? ? ≥ 0. Since B V ≤ 0, we have ei → 0, (i = 1, 2, 3) i.e., limt→∞ ei = 0. This completes the proof. 4. Numerical simulations In this section, numerical simulations are pre- sented to verify the e?ectiveness of the proposed syn- chronization controller. The parameters are chosen to be a = 0.2, b = 0.5, c = 5.7 and delay parameters are chosen as τ1 = 1, τ2 = 2. So that the double-delayed R¨ ossler system exhibits chaotic behaviour if no con- trol is applied to it. The initial conditions of the drive system are x1(0) = ?4, x2(0) = 4, and x3(0) = 0, and those of the response system are y1(0) = 1, y2(0) = 1, and y3(0) = 1. Moreover, the initial values of the estimated parameters are chosen to be ? a(0) = 0.1, ? b(0) = 0.1, and ? c(0) = 0.1. Furthermore, the control gains are chosen as k1 = ?3, k2 = ?2.5. The simulation results are shown in Figs. 2C4. The evolution of the errors is depicted in Fig. 2, which displays the time response of the errors e →

0 with t → ∞. Figure

3 shows the portrait of state vari- ables between the drive and response systems. Figure

4 depicts the evolution of the estimated parameters of the double-delayed R¨ ossler system, which shows that the estimates of the unknown parameters adapt them- selves to the true values. These results show that the required synchronization has been achieved by our de- signed control law (7) and parameter update law (8). 080507-3 Chin. Phys. B Vol. 20, No.

8 (2011)

080507 Fig. 2. The time evolution of errors between the drive and response systems. Fig. 3. Plots of state variables between drive and response systems. Fig. 4. The time evolution of the estimated unknown parameters. 5. Conclusion This paper investigated the synchronization of double-delayed R¨ ossler systems. A simple but e?cient controller was designed to synchronize the chaotic dy- namics between two coupled identical double-delayed R¨ ossler chaotic systems, which could be proved by the Lyapunov stability theorem. By virtue of the synchronization-based method, the unknown param........