Page 433 - Mathematical Techniques of Fractional Order Systems

418 Mathematical Techniques of Fractional Order Systems

where x is the savings of households, y is the gross domestic product (GDP),
and z is the foreign capital inflow. The system parameters m, p, dc, s, and r
are the marginal propensity, the ratio of capitalized profit, the value of the
potential GDP, the between output and capital, the ratio between capital
inflow and savings, and the ratio between debt refund and output, respec-
tively. The external active control inputs are U i , i 5 1; 2; 3 were designed to
ensure that the system is asymptotically stable at one of its three equilibrium
points. When the systems parameters are: m 5 0:02, p 5 0:4, c 5 50, d 5 1,
and s 5 10, the equilibrium points are: E 0 5 ð0; 0; 0Þ,
r 5 0:1 ,
4
E 1;2 5 ð 6 0:024; 6 2:4; 6 4:8 3 10 Þ. To stabilize the system at E 0 , the con-
trol inputs must be:
0 1 0 2 1
U 1 21:4x 1 0:98y 1 0:4xy
5 ;
U 2
@ A @ 2y 2 49:9z A ð14:27Þ
210x 2 z
U 3
and to stabilize it at E 1;2 , the control inputs should be:
0 1 0 2 2 1
U 1 0:904x 1 1:02608y70:002304 1 0:4xy 6 1:92xy 6 0:0096y
U 2
@ A 5 @ 2y 2 49:9z A :
U 3 210x 2 z
ð14:28Þ
Linear feedback control technique was used by El-Sayed et al. (2016)to
achieve stability of a novel hyperchaotic model. The controlled system struc-
ture was given as (El-Sayed et al., 2016):
α
2
D x 5 a 1 x 1 b 1 y 1 c 1 xw 2 k 1 ðx 2 x Þ; ð14:29aÞ

α

D y 5 a 2 x 1 b 2 y 1 h 1 z 1 c 2 w 2 k 2 ðy 2 y Þ; ð14:29bÞ
α
D z 5 b 3 y 2 k 3 ðz 2 z Þ; ð14:29cÞ

α

D w 5 a 3 x 1 h 2 z 2 k 4 ðw 2 w Þ; ð14:29dÞ
where k i $ 0 and i 5 1; 2; 3; 4 are the feedback control gains and

ðx ; y ; z ; w Þ is an equilibrium point of the system. It was seen that the sys-

tem is controlled only in the fractional order (El-Sayed et al., 2016). At para-
meters values: a 1 5 5:9, c 1 5 19, a 2 5 7:82, b 1 5 1:36, b 2 5 1:5, h 1 5 8:5,
c 2 5 2:7, b 3 5 7:8125, a 3 5 11:6, h 2 5 5:731, and control gains k 1 5 22:2288,
k 2 5 k 3 5 k 4 5 0, the system has two symmetric equilibrium points:
E 1;2 5 ð 6 0:1603; 0; 6 0:3245; 6 0:5573Þ (El-Sayed et al., 2016). Fig. 14.4
shows the system convergence to these equilibrium points at α 5 0:95 which
is practically achieved in under 8 seconds.
14.4 SYNCHRONIZATION
Chaotic synchronization can be defined as the state when two or more cha-
otic systems adjust their responses or some aspect of them as a result of