Page 337 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 11 327

11.6 SYSTEM DESCRIPTION
The fractional order new hyperchaotic system (Gao et al., 2007) is given by

8 q
d x 1
>
> 5 a 1 ðy 1 2 x 1 Þ
dt
> q
>
>
>
>
q
>
d y 1
>
>
>
> q 5 dx 1 2 x 1 z 1 1 cy 1 2 w 1
dt
>
<
q ð11:14Þ
d z 1
>
> 5 x 1 y 1 2 bz 1
> q
> dt
>
>
>
> q
d w 1
>
>
> 5 x 1 1 k
>
> q
: dt
where x 1 ; y 1 ; z 1 ; w 1 are the state variables, q is the fractional order satisfying
0 , q , 1 and a; b; c; d; k are the parameters.
Applying the discretization scheme, it was found that hyperchaos indeed
exists in the new system with fractional order. The parameters are always
chosen as a 5 36; b 5 3; c 5 28; d 5 16 and k 5 0:5. It was demonstrated
that hyperchaos does exist in the fractional order system with order less than
4. It was found that when 0:72 , q , 1, the fractional order system (11.14)
displays hyperchaotic behaviors. For example, when q 5 0:9 and q 5 0:72,
hyperchaotic attractors are found and the phase portraits are shown in
Figs. 11.2 and 11.3, respectively. We calculated the two largest Lyapunov
exponents of this system using the well-known Wolf algorithm. The values
of the two largest Lyapunov exponents are λ 1 5 12:3014 and λ 2 5 0:2318
when q 5 0:9. The values of the two largest Lyapunov exponents are
λ 1 5 8:2130 and λ 2 5 0:1015 when q 5 0:72. Obviously, the fractional order
system (11.14) has hyperchaos. Here we have considered the parameters
values a 5 36; b 5 3; c 5 28; d 52 16; k 5 0:5 and q 5 0:95, the phase
portraits of the system showing hyperchaotic behavior is shown in Figs. 11.2
and 11.3.
The fractional order hyperchaotic Gao system (Gao et al., 2015)is
given by
8 q
d x 2
>
> 52 a 1 x 2 1 a 2 y 2
dt
> q
>
>
>
q
>
>
d y 2
>
>
>
> 5 a 3 x 2 2 x 2 z 2 2 y 2 1 w 2
dt
> q
<
q ð11:15Þ
d z 2 2
>
> 5 x 2 a 4 ðx 2 1 z 2 Þ
> q 2
> dt
>
>
>
> q
>
d w 2
>
>
> 52 a 5 x 2
dt
> q
: