Page 672 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 643

th
GMðx i Þ is generalized i system with memory, PLðx i Þ is percolation lattice
th
i system, and η ðx i Þ is the fractional part map of sawtooth wave, verifying:
x
p
ηðx i Þ:D 3A 3 frac 1 φ
t
T
where A is the amplitude, T is the period of the wave, φ is its phase, and
frac ðx i Þ is the fractional part, verifying:
frac ðxÞ 5 x 2 ½xŠ
The considered procedure needs the implementation of the following iter-
ative algorithm:
Step 1: Initialization of the two fractional orders chaotic systems;
Step 2: Simulation of the considered two fractional orders chaotic sys-
tems according to the algorithm;
^ N
Step 3: Initialization of the master’s X t 5 fg system related to the
α
x n n 5 0
considered two fractional orders chaotic systems;
Step 4: Using the proposed method to achieve the behavior of synchroni-
zation between two new fractional orders chaotic systems;
^
Step 5: Determined the corresponding slave system Y t 5 α N
y n
n 5 0
defined as:
^
^
^
Y t 5 X 1 SðXÞ; YAU
where SðXÞ is the synchronization of algorithm with period T in its phase φ
and U is the m-dimensional input vector that will be used.
Step 6: Start the iterative algorithm for topological synchronization using
the master slave schema;
Step 7: Using the iterative procedure to evaluate the effect of the “prox-
imity” capture of the average Poincare ´ return time as a criterion of chaos
control of topological synchronization systems with while tracking control of
chaotic nonlinear fractional order systems;
p ^
Step 8: Comparing and the Poincare ´ recurrence diagrams for both D t X,
p ^
D t Y systems and respectively interpreting their related topological struc-
tures. Then, define the joint Poincare ´ recurrence diagram for these coupled
systems which are essentially different and the related topological
measures of its quantification analysis. Mathematically, a new index will
be defined and which is based on the average probability of their joint
recurrence over time, it is considered as criterion for the detection of
topological synchronization in master slave dynamical system, and it is
given as:
1 X N X N
x
x; y Θ ε 2:x i 2x j :
R ij 5 i;j51; 2;...;N
N i51 j51
y
Θ ε 2:y i 2y j : ; x; yAU

i;j51; 2;...;N