# Adaptive Self-Scaling Brain-Storm Optimization via a Chaotic Search Mechanism
## 3 Chaotic Maps  
One-dimensional irreversible mapping is the simplest system capable of producing chaotic motion. The chaotic sequences produced by different chaotic maps, as well as their distributions and ergodic properties, are completely different. Due to the ergodicity and non-repetition of chaotic sequences, searches with various chaotic maps outperform searches with a single chaotic map in terms of both the search range and the accuracy. Chaotic maps are used to generate chaotic sequences that are then implemented in the CLS strategy. In this study, 12 one-dimensional chaotic maps were selected to form chaotic mutation operators to constitute the search strategies, which effectively prevented the algorithm from falling into local optima. The 12 chaotic maps are briefly described as follows:A logistic map is a classic chaotic map in the nonlinear dynamics of a biological population that can be expressed as follows: \[y_{n+1}=\beta y_{n}(1-y_{n}),\] (3) where \(y_{n}\) denotes the \(n\)-th chaotic sequence and \(y_{n}\in(0,1)\), \(y_{0}\
ot\in\{0,0.25,0.5,0.75,1\}\). In this study, we set \(\beta=4\), and the initial chaotic value was \(y_{0}=0.152\). 2. A piecewise linear chaotic map (PWLCM) has a constant density function in the defined interval. The simplest PWLCM can be obtained by the following equation: \[y_{n+1}=\left\{ \begin{array}{cl}y_{n}/\beta,&y_{n}\in(0,\beta)\\ (1-y_{n})(1-\beta),&y_{n}\in[\beta,1)\end{array} \right.\] (4) In our experiment, the initial chaotic sequence was \(y_{0}=0.002\), and we set \(\beta=0.7\). 3. A Singer map is similar to a one-dimensional chaotic system: \[y_{n+1}=\beta\Big{(}7.86y_{n}-23.31y_{n}^{2}+28.75y_{n}^{3}-13.302875y_{n}^{4} \Big{)},\] (5) when the Singer map shows chaotic behavior, parameter \(\beta\in(0.9,1.08)\). In this study, we set \(y_{0}=0.152\) and \(\beta=1.073\). 4. A sine map is similar to a unimodal logistic map and can be expressed by the following formula: \[y_{n+1}=\frac{\beta}{4}\sin(\pi y_{n}),\] (6) where \(\beta\in(0,4]\), \(y_{0}\in(0,1)\), and we determined that \(\beta=4\) and \(y_{0}=0.152\). 5. A circle map is a simplified one-dimensional model for driven mechanical rotors and phase locked loops in electronics that maps a circle onto itself and can be expressed by the following equation: \[y_{n+1}=y_{n}+\alpha-\frac{\beta}{2\pi}\sin(2\pi y_{n})\ \mathrm{mod}\ (1),\] (7) when \(\alpha=0.5\) and \(\beta=2.2\), the chaotic sequences within the interval of (0,1) can be obtained based on the above equation. The initial value was set to \(y_{0}=0.152\). 6. A Bernoulli shift map is a type of piecewise linear map that is similar to a tent map and can be expressed as follows: \[y_{n+1}=\left\{ \begin{array}{cl}y_{n}/(1-\beta),&0<y_{n}\leq 1-\beta\\ (y_{n}-1+\beta)/\beta,&1-\beta<y_{n}<1\end{array} \right.\] (8) In this study, we set \(\beta=0.4\) and \(y_{0}=0.152\). 7. An iterative chaotic map with infinite collapses (ICMIC) obtains infinite fixed points based on the following equation: \[y_{n+1}=\sin\frac{b}{y_{n}},\] (9) where the adjustable parameter \(b\in(0,+\infty)\). In this study, \(b=70\). Therefore, the chaotic sequences are in \((-1,\,1)\), and \(y_{n}\
eq 0\); if the generated sequence is negative, the absolute value is taken. 8. A sinusoidal map can be generated by the following formula: \[y_{n+1}=by_{n}^{2}\sin(\pi y_{n}),\] (10) where \(y_{0}=0.74\), and \(b=2.3\). 9. Chebyshev maps are widely used in digital communication, neural computing and safety maintenance. The process to generate chaotic sequences is defined as follows: \[y_{n+1}=\cos\Big{(}\rho\cdot\cos^{-1}y_{n}\Big{)},\] (11)where \(\rho=5\), and \(y_{0}=0.152\). * A topological conjugacy relationship exists between the famous logistic map and the tent map, so the two are interconvertible. Tent map sequences can be calculated by the following formula: \[y_{n+1}=\left\{ \begin{array}{cc}y_{n}/\beta,&0<y_{n}\leq\beta\\ (1-y_{n})/(1-\beta),&\beta<y_{n}\leq 1\end{array} \right.\] (12) where \(\beta=4\), and \(y_{0}=0.152\).