Page 371 - Mathematical Techniques of Fractional Order Systems

362 Mathematical Techniques of Fractional Order Systems

12.3.2 Dual Combination Synchronization With Order m . n
The fractional order Lu hyperchaotic system and the fractional order 4D
Integral order hyperchaotic system are taken as the first two master
systems as
α
d x 11
dt α 5 a 11 ðx 12 2 x 11 Þ 1 x 14
α
d x 12
dt α 52 x 11 x 13 1 a 13 x 12
α ð12:17Þ
d x 13
dt α 5 x 11 x 12 2 a 12 x 13
α
d x 14
dt α 5 x 11 x 13 1 a 14 x 14

and
α
d y 11
dt α 5 b 11 y 11 2 y 12
α
d y 12 5 y 11 2 y 12 y 2
dt α 13
α ð12:18Þ
d y 13 52 b 12 y 12 2 b 13 y 13 2 b 14 y 14
dt α
α
d y 14 5 y 13 1 b 15 y 14 ;
dt α

where x 1i ; y 1i ði 5 1; 2; 3; 4Þ are states variables and
a 1i ; ði 5 1; 2; 3; 4Þ, b 1i ði 5 1; 2; 3; 4; 5Þ are the constant parameters.
The fractional order Chen hyperchaotic system (Matouk and Elsadany,
2014) and fractional order Lorenz hyperchaotic system (Chen et al., 2011)
are considered as
α
d x 21
dt α 5 a 21 ðx 22 2 x 21 Þ 1 x 24
α
d x 22
dt α 5 a 22 x 21 2 x 21 x 23 1 a 23 x 22
α ð12:19Þ
d x 23 5 x 21 x 22 2 a 24 x 23
dt α
α
d x 24 5 x 22 x 23 1 a 25 x 24 ;
dt α

where x 21 ; x 22 ; x 23 and x 24 are states variables and a 21 ; a 22 ; a 23 ; a 24 , and
a 25 are constant parameters. The phase portraits of (12.19) in x 21 2 x 22 2 x 23 ,