Page 685 - Mathematical Techniques of Fractional Order Systems

656 Mathematical Techniques of Fractional Order Systems

FIGURE 21.21 Simulation result of recurrence diagram of the master slave related to two
considered fractional order Ro ¨ssler’s systems.

Step 8: The Poincare ´ recurrence diagram and fractal dimensions for mas-
ter slave system are determined. After executing the iterative algorithm for
topological synchronization for two iterations, the recurrence diagram for
master slave joint system is in the phase coherent. Then, it may be synchro-
nized as visualized in Fig. 21.21. For these two iterations, their respectively
related fractal dimensions are D 1 5 1.619 and D 25 1.401 and their respec-
tively related averages Poincare ´ returns times are τ 1 5 3.8452 and
τ 2 5 3.8192.
Step 9: As the averages Poincare ´ returns times are determined, then the
effect of the “proximity” is computed as:
ε 5 τ 1 2 τ 2 5 0:026

Step 10: Define the Lyapunov stability.

21.7 CONCLUSION

To conclude, the synchronization of complex systems that has been widely
studied in the last two decades has several applications in various fields.
Now, synchronization analysis and recurrence in fractional order chaotic sys-
tems has become an open and exciting axis of research. In particular, topo-
logical synchronization by means of recurrences for some fractional order
chaotic systems is discussed in this chapter. The results show that the control