Page 412 - Mathematical Techniques of Fractional Order Systems

On the Fractional Order Generalized Discrete Maps Chapter | 13 399

introduced in the fourth row. The last two rows present 3D snapshots of the
complete bifurcation diagrams for both maps.

13.3.4 Generalized Fractional Logistic Map Design

Using the generalization parameters ða; bÞ introduced as well as the FO
parameter α, the chaotic range of the map can be fully controlled to
achieve a certain bifurcation diagram, allowing different map designs. The
function value at which bifurcation takes place x b at ρ can be specified; as
b
well as x max at ρ max ;or x min ,at ρ min , in both sides positive and negative
bifurcations, and hence calculate the values of the generalization para-
meters ða; bÞ, using the equations derived in Table 13.5, to achieve the spe-
cifications required.
Four design cases are introduced and summarized in Table 13.7, with the
design specifications, the corresponding bifurcation diagrams, as well as the
Lyapunov exponent graph verifying the chaotic region.

13.4 FRACTIONAL ORDER GENERALIZED TENT MAP
This section focuses on the Wu definition (Wu and Baleanu, 2014) of the
fractional logistic map and extends it to present the FO tent map depending
on the following theorem.

Theorem 1: For the delta fractional difference equation (Chen et al., 2011)

C ν
Δ uðtÞ 5 ft 1 ν 2 1; uðt 1 ν 2 1ÞÞ; ð13:26Þ
ð
a
k
Δ uðaÞ 5 u k ; m 5 ½νŠ 1 1; k 5 0; ...; m 2 1; ð13:27Þ
the equivalent discrete integral equation can be obtained as
t2ν
1 X
uðtÞ5u 0 ðtÞ1 ðt2σðsÞÞ ðν21Þ fðs1ν 21;uðs1ν 21ÞÞ; tAN a1m ;
ΓðνÞ
s5a1m2ν
ð13:28Þ
where the initial iteration u 0 ðtÞ reads

m21 ðkÞ
X ðt2aÞ k
u 0 ðtÞ 5 Δ uðaÞ: ð13:29Þ
k!
k50