Page 435 - Mathematical Techniques of Fractional Order Systems

420 Mathematical Techniques of Fractional Order Systems

Phase synchronization: Happens when the phases of both systems are
locked while the dependency of the amplitudes is weak.
Lag synchronization: When outputs of both systems are identical except
for a constant time shift.
Antisynchronization: Is similar to complete synchronization except the
amplitudes are additive inverses to each other (Srivastava et al., 2014).
Projective synchronization: When both responses are the same except for
a constant factor (Jiang et al., 2017; Ouannas et al., 2017i,h).
Impulsive synchronization: When the coupling is forced to be on/off
according to some condition.
Incomplete synchronization: When the difference between the responses
of the synchronized systems occasionally differs from zero.
Complete desynchronization: It occurs when both systems differ in every
detail.

Another important form of synchronization is between two different sys-
tems: different dimensions (Ouannas et al., 2017k,f) or even integer and frac-
tional (Ouannas et al., 2017e). The following subsections discuss three
examples of synchronization of FOCS.

14.4.1 Synchronization of Fractional Order Modified
Van der Pol-Duffing Circuit

The system is given as (Matouk, 2011):
α
3
D x 52 vðx 2 μx 2 yÞ; ð14:30aÞ
α
D y 5 x 2 γy 2 z; ð14:30bÞ
α
D z 5 βy; ð14:30cÞ
where the original integer order system ðα 5 1Þ exhibits chaotic behavior at
parameter values: β 5 200, μ 5 0:1, v 5 100 , and γ 5 1:6. The system has
three equilibrium points: E 0 5 ð0; 0; 0Þ and E 1;2 5 ð 6 μ; 0; 6 μÞ. The equilib-
rium point E 0 is a saddle point of index 1 while the other two equilibrium
points E 1;2 are saddle points of index 2. If the eigenvalues at any equilibrium
point are given (in general) by the polynomial:
2
3
PðλÞ 5 λ 1 a 1 λ 1 a 2 λ 1 a 3 5 0; ð14:31Þ
then, the discriminant is given as (Matouk, 2011):
3
2
3
2
DðPÞ 5 18a 1 a 2 a 3 1 ða 1 a 2 Þ 2 4a 3 ða 1 Þ 2 4ða 2 Þ 2 27ða 3 Þ : ð14:32Þ
Based on the discriminant, the authors derived the fractional
Routh Hurwitz conditions for stability then they proposed two schemes for
synchronization of this system: unidirectional linear error feedback coupling