Table 2 Chaotic maps.

From: Improved optimization based on parrot’s chaotic optimizer for solving complex problems in engineering and medical image segmentation

Map name

Functions

Range

Tchebychev map

\({x_{k+1}}=\cos (k{\cos ^{ - 1}}({x_k}))\)

(− 1,1)

Circular map

\({x_{k+1}}={x_k}+b - (\frac{a}{{2\pi }})\sin (2\pi {x_k})\bmod (1)\)

(0,1)

Gauss map

\({x_{k+1}}=\left\{ {\begin{array}{*{20}{c}} {0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_k}=0} \\ {\frac{1}{{{x_k}}}\,\,\bmod (1)\,\,\,\,\,\,{\text{otherwise}}\,\,\,\,\,} \end{array}} \right.\)

(0,1)

Iterative map

\({x_{k+1}}=\sin (\frac{{a\pi }}{{{x_k}}}),\,\,\,\,\,a=0.7\)

(− 1,1)

Logistics map

\({x_{k+1}}=a{x_k}(1 - {x_k}),\,\,\,\,\,a=4\)

(0,1)

Piecewise map

\({x_{k+1}}=\left\{ {\begin{array}{*{20}{c}} {\frac{{{x_k}}}{p}\,\,\,\,\,0 \leqslant {x_k} \leqslant p} \\ {\frac{{{x_k} - p}}{{0.5 - p}}\,\,\,\,\,P \leqslant {x_k} \leqslant 1/2} \\ {\frac{{1 - p - {x_k}}}{{0.5 - p}}\,\,\,\,\,1/2 \leqslant {x_k} \leqslant 1 - p} \\ {\frac{{1 - {x_k}}}{p}\,\,\,\,\,1 - p \leqslant {x_k} \leqslant 1} \end{array}} \right.\)

(0,1)

Sine map

\({x_{k+1}}=\frac{a}{4}\sin (\pi {x_k}),\,\,\,\,\,\,\,a=4\)

(0,1)

Singer map

\({x_{k+1}}=\mu (7.86{x_k} - 23.31x_{k}^{2}+28.75x_{k}^{3} - 13.302875x_{k}^{4}),\,\,\,\,\,\,\,\mu =1.07\)

(0,1)

Sinusoidal map

\({x_{k+1}}=ax_{k}^{2}\sin (\pi {x_k}),\,\,\,\,\,\,\,a=2.3\)

(0,1)

Tent map

\({x_{k+1}}=\left\{ {\begin{array}{*{20}{c}} {\frac{{{x_k}}}{{0.7}},\,\,\,\,\,\,{x_k} \prec 0.7} \\ {\frac{{10}}{3}(1 - {x_k}),\,\,\,\,\,{x_k} \geqslant 0.7} \end{array}} \right.\)

(0,1)