4) The Piecewise Linear Chaotic Map (PLCM)
The PLCM [23] is defined as follows, in Eq. (6):
xkþ 1ðÞ¼fxkðÞ; pðÞ¼
xkðÞ=p; 0≤ xkðÞ< p;
xkðÞ−pðÞ= 0:5−pðÞ; p≤ xkðÞ< 0:5;
1−p−xkðÞðÞ= 0:5−pðÞ; 0:5 ≤ xkðÞ< 1−p;
1−xkðÞðÞ=p; 1−p≤xkðÞ< 1;
8
>
>
<
>
>
:
ð6Þ
wh
ere p is the control parameter and satisfies 0 < p < 0.5, this map shows good performance.
2.1.3 Selection of chaotic algorithm based on roulette wheel
The roulette wheel selection algorithm, called proportional selection algorithm, is
mainly used in the optimization algorithm [31 , 37, 46]. In this paper, we use t his
method to select the current chaotic dynamical system. The specific algorithm is as
follows:
Algorithm 1: roulette wheel selection [F
k
←selection(p
1
, p
2
, p
3
, ..., p
n ;
F
1
, F
2
, ..., F
n
)]
Suppose a given set of chaotic dynamic systems is F
1
, F
2
, ..., F
n
with
corresponding individual selection probability p
1
, p
2
, p
3
, ..., p
n
. Then
1) Calculate individual cumulative probability
1
[1, ]
i
ii
j
qpin
;
2) Generate a uniformly distributed random number r (see 2.1.4), between 0 and 1;
3) If r≤q
1
, then chaotic system F
1
is chosen;
4) If q
k-1
<r≤q
k
(2≤k≤n), then chaotic system F
k
is chosen.
2.1.4 Pseudorandom number generator
In section 2.1.3, random numbers are needed. In our algorithm, the PLCM is used to finish this
task. Usually, random numbers should have randomness and uniform distribution.The PLCM,
like any other chaotic system, has natural randomness. More importantly, its state variables
have a uniform distribution property. Proof is as follow.
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