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Advances, Systems and Applications

Table 6 Chaotic maps

From: An efficient population-based multi-objective task scheduling approach in fog computing systems

Number Name Chaotic Map Diagram
1 Chebyshev [47] \(\begin {array}{c}x_{(i+1)}= \cos (i cos^{-1} x_i) \\ x_0=0.7\end {array}\)
2 Circle [48] \(\begin {array}{c} x_{(i+1)}=(x_i+b-(\frac {a}{2\pi })\sin (2 \pi x_i))\text {mod}(1) \\ a=0.5,b=0.2,x_0=0.7 \end {array}\)
3 Cubic [49] \(\begin {array}{c} x_{(i+1)}=2.59 x_i (1 - x_{i}^{2}) \\ x_0 = 0.7 \end {array}\)
4 Guass [50] \(\begin {array}{c} x_{(i+1)} = \left \{ \begin {array}{lll} 0 & \qquad \text {if } x_i = 0 \\ \frac {1}{x_i} \text {mod}(1) &\qquad \text {if } x_{i}\neq 0 \end {array} \right. \\ x_0=0.7 \end {array}\)
5 Iterative [51] \(\begin {array}{c} x_{(i+1)} = \sin (\frac {a\pi }{x_i}) \\ a=0.7, x_0=0.7 \end {array}\)
6 Logistic [51] \(\begin {array}{c} x_{(i+1)}=a x_i (1 - x_{i}) \\ a=4, x_0=0.7 \end {array}\)
7 Piecewise [52] \(\begin {array}{c} x_{(i+1)} = \left \{\begin {array}{lll} \nicefrac {x_i}{p} & \qquad \text {if } x_i < p \\ \nicefrac {x_i - p}{0.5 - p} &\qquad \text {if } p\leq x_{i}\leq 0.5 \\ \nicefrac {1 - p - x_i}{0.5 - p}&\qquad \text {if } 0.5 \leq x_{i}\leq 1-p \\ \nicefrac {1 - x_i}{p}&\qquad \text {if } 1-p < x_i \end {array} \right. \\ p = 0.4, x_0=0.7 \end {array}\)
8 Sine [53] \(\begin {array}{c} x_{(i+1)}= \frac {a}{4} \sin (\pi x_i) \\ a=0.4, x_0 = 0.7 \end {array}\)
9 Singer [54] \(\begin {array}{c} x_{(i+1)}= \mu (7.86 x_i - 23.31 x_{i}^{2} + 28.75 x_{i}^{3} - 13.302875 x_{i}^{4}) \\ \mu = 1.073, x_0=0.7 \end {array}\)
10 Sinusoidal [55] \(\begin {array}{c} x_{(i+1)}=a x_{i}{2}\sin (\pi x_i) \\ a=2.3, x_0 = 0.7 \end {array}\)
11 Tent [56] \(\begin {array}{c} x_{(i+1)} = \left \{ \begin {array}{lll} \nicefrac {x_i}{0.7} & \qquad \text {if } x_i < 0.7 \\ \nicefrac {10}{3} (1-x_i) &\qquad \text {otherwise } \\ \end {array} \right. \end {array}\)