Page 684 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 655

21.6.2.2 Algorithm of Topological Synchronization of Fractional
Order Ro ¨ ssler’s Systems
Step 1: Let consider two fractional order Ro ¨ssler’s systems as (21.21).
The first system is given for parameters (a 5 0.5, b 5 0.2, c 5 10), orders
p 1 5 p 2 5 p 3 5 0.95 and initial conditions (x 0 520.5, y 0 5 0, z 0 5 1) which
can be topologically synchronized with the second system given for para-
meters (a 5 2.5, b 5 5, c 5 4), orders p 1 5 p 2 5 p 3 5 0.95 and initial condi-
tions (x 0 5 1, y 0 5 1, z 0 5 1).
Step 2: Both systems are simulated according to previous the algorithm
developed by Vladimirsky and Ismailov (2015a,b) and respectively repre-
sented as it was previously visualized.
^ N
Step 3: Let X t 5 fg be associated observable two considered frac-
α
x n n 5 0
tional order Ro ¨ssler’s systems.
Step 4: In order to determine the behavior of synchronization between
two new chaotic systems by using the proposed method, it is supposed that
^
X t is the master.
^
Step 5: The corresponding slave system Y t 5 α N n 5 0 is determined.
y n
Step 6: The iterative algorithm for topological synchronization on mas-
ter slave system with while tracking control for chaotic nonlinear fractional
order systems is executed.
Step 7: We consider the average Poincare ´ return time as a criterion for
chaotic topological synchronization systems with while tracking control. The
first-return map of the master slave system is ported in Fig. 21.20.

FIGURE 21.20 Simulation result of first-return map of the master slave related to two consid-
ered fractional order Ro ¨ssler’s systems in plane.