Page 682 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 653

x y z t t t
20
Series of fractional-order Rössler’s system 10 5 0
15

–5
–10
0 20 40 60 80 100
Time
FIGURE 21.15 Simulation result of all fractional order Ro ¨ssler’s system (21.21) variables ver-
sus time 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 condi-
tions (x 0 520.5, y 0 5 0, z 0 5 1).

FIGURE 21.16 Simulation result of recurrence diagram of the fractional order Ro ¨ssler’s sys-
tem (21.21) in i j plane 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).

in Fig. 21.14. All model variables are represented versus time in Fig. 21.15.
The related recurrence diagram is ported in Fig. 21.16.
In a second case of the same fractional order Ro ¨ssler’s system (21.21)
with even values of the commensurate orders p 1 5 p 2 5 p 3 5 p 5 0.95, if
parameters are fixed as (a 5 2.5, b 5 5, c 5 4), initial conditions as (x 0 5 1,
y 0 5 1, z 0 5 1), and computational time 100s for time step h 5 0.01, then the
fractional order Ro ¨ssler’s system (21.21) exhibits a strange attractors and the
chaotic butterfly-effect which are given in Fig. 21.17. All model variables
are represented versus time in Fig. 21.18. The related recurrence diagram is
depicted in Fig. 21.19.