Page 469 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 15 455

been investigated using active sliding mode controller (Tavazoei and Haeri,
2008), a modified Duffing system with excited parameters (Ge and Ou,
2008), and via active control (He and Luo, 2012).

15.3.2 Arenodo System
Arneodo et al. (1982) proposed a third order differential equation for the
thermohaline convection given as

dx 1
5 x 2
dt
dx 2
5 x 3 ð15:7Þ
dt
dx 3 3
52 β x 1 2 β x 2 2 β x 3 1 β x
3
4 1
1
2
dt
where β , β , β , and β are constant parameters. The integer order form of
1 2 3 4
the Arneodo system was found to have three unstable equilibrium points and
positive Lyapunov exponent (Motallebzadeh et al., 2009).
The fractional order Arneodo system was introduced by Lu (2005) and
described as
q 1
d x 1
5 x 2
dt
q 2
d x 2
5 x 3 ð15:8Þ
dt
q 3
d x 3 3
52 β x 1 2 β x 2 2 β x 3 1 β x
3
2
1
4 1
dt
where q is the fractional order satisfying 0 , q # 1. The fractional order
Arneodo was found to have a maximum Lyapunov exponent of 0.22 when
β 5 0:4 and q 5 0:9(Lu, 2005). The phase portrait of the fractional order
3
Arneodo system is shown in Fig. 15.2. In this study, β 52 5:5, β 5 3:5,
1 2
β 5 0:4, β 52 1, q 1 5 q 2 5 q 3 5 0:9.
3 4
The Arenodo chaotic system has been the subject of different research, such
as control of chaos in the integer order Arneodo system (Motallebzadeh et al.,
2009), synchronization of the fractional order Arenodo system (Hajipour and
Aminabadi, 2016; Lu, 2005), backstepping fuzzy adaptive control synchroniza-
tion (Wang and Fan, 2015), and FPGA implementation (Shah et al., 2017).