Page 667 - Mathematical Techniques of Fractional Order Systems

638 Mathematical Techniques of Fractional Order Systems

21 P N P N
ð
½
log 2 NN21ފ i51 j5i11 R ij
D c 5 lim
ε-0 log ðεÞ
where ε is the sphere of radius ε, N is the number of considered condition x i
and R ij is the expression as defined in (21.1).

21.4.3 The Maximum Lyapunov Exponent

This measure starts from the idea that two points are initially very close in
data space set have trajectories divergent exponentially. This indicator was
proposed by Wolf et al. (1985); therefore it makes possible to know whether
there is sensitivity to the initial conditions in the data. The exponent of
Lyapunov is expressed as follows:

Definition: The Lyapunov exponent is a measure of divergence rate of two
trajectories after an infinitesimal deviation
1
λ max 5 lim log λ T

i
T-N T
where for all i, λ i are all eigenvalues of the Jacobian matrix J evaluated at
the equilibrium points corresponding to the considered fractional order cha-
otic system.

The maximum Lyapunov exponent estimation value is interpreted as
flows:
If λ max , 0: it is a converged dynamical system to a stable fixed point.
If λ max 5 0: it is a limit cycle, the dynamical system is stable in sense of
Lyapunov.
If λ max . 0: it is an instable dynamical system with a chaotic behavior.
If λ max 5 N: it is a noise.
Recently, using the fractional order Lyapunov direct method, according
to Aguila-Camacho et al. (2014), the Lyapunov function is given in the qua-
dratic form as:
1
VðX t Þ 5 Ω Ω p ð21:17Þ
0
2 p
where X t 5 (x 1t , x 2t ,..., x nt )’, p is a rational number between 0 and 1 and Ω p
is the memory function of order p. According to Chen et al. (2014) the frac-
tional order Lyapunov stability using Lyapunov direct method has been
developed by the following theorem.