Page 437 - Mathematical Techniques of Fractional Order Systems

422 Mathematical Techniques of Fractional Order Systems

β
D YðτÞ 5 bXðτÞ; ð14:38bÞ
where XðtÞ 5 vðtÞ ; τ 5 t ; YðtÞ 5 Ri L ðtÞ and single membrane conductance is
V R RC m V R
defined as:

vðτÞ vðτÞ
σ m 5 f ðvðτÞÞ 5 1 2 1 2 ; ð14:39Þ
V T V p
where V T and V p are the membrane threshold and peak voltages, respectively.
When synchronizing two coupled neurons, their combined system is
given by:
α
D X 1 ðtÞ 5 X 1 ðtÞðX 1 ðtÞ 2 1Þð1 2 rX 1 ðtÞÞ 2 Y 1 ðtÞ 1 I 0 ðtÞ 2 gðX 1 ðtÞ 2 X 2 ðtÞÞ;
ð14:40aÞ
β
D Y 1 ðtÞ 5 bX 1 ðtÞ; ð14:40bÞ
α
D X 2 ðtÞ 5 X 2 ðtÞðX 2 ðtÞ 2 1Þð1 2 rX 2 ðtÞÞ 2 Y 2 ðtÞ 1 I 0 ðtÞ 2 gðX 2 ðtÞ 2 X 1 ðtÞÞ;
ð14:40cÞ
β
D Y 2 ðtÞ 5 bX 2 ðtÞ; ð14:40dÞ
where g is the control gain, r 5 V p and I 0 ðtÞ is the external stimulation. The
V T
dynamical error system is given as:
α
2
2

D e x 5 ð1 1 rÞðX 1 1 X 2 Þ 2 rðX 1 X 1 X 1 X 2 Þ e x 2 e y ; ð14:41aÞ
1 2
α
D e y 5 be x : ð14:69Þ
The authors also discussed the generalized system of N coupled neurons.
It can be seen from their simulations that the synchronization is faster as the
fractional order decreases and also as the control gain g approaches 0:5 the
synchronization is also faster.
14.4.3 Sliding Mode Synchronization of Uncertain
Chaotic Systems
Synchronization of a FOCS which involves model uncertainty is a rather dif-
ficult task. Hosseinnia et al. (2010) discussed this issue with an example on
the Duffing Holmes system. The master system is given as:
q
D x 1 5 x 2 ; ð14:42aÞ
3
q
D x 2 5 x 1 2 αx 2 2 x 1 βcosðtÞ; ð14:42bÞ
1
while the slave system is given by:
q
D y 1 5 y 2 ; ð14:43aÞ