*3.2. Chaotic Mapping*

Chaotic is a deterministic stochastic method found in non-periodic, non-convergence and bounded nonlinear dynamic systems. Mathematically, chaotic is the randomness of a simple deterministic dynamic system, and a chaotic system is considered as the source of randomness. The essence of chaotic is obviously random and unpredictable, and it also has regularity [59].

As an important part of the population initialization algorithm, its result directly affects the convergence speed and quality of the algorithm [60,61]. For example, uniform distribution has more complete coverage of solution space than random distribution, and it is easier to obtain good initial solutions. A classical HHO algorithm uses random population initialization operation, which cannot cover the whole solution space. A chaotic sequence has ergodicity, randomness, and regularity in a certain range. Compared with random search, chaotic sequence searches the search space thoroughly with higher probability, which enables the algorithm to go beyond the local optimum and maintain the diversity of the population. Based on the above analysis, to obtain a good initial solution position and speed up the convergence of the population, seven common chaotic mappings Sinusoidal, Tent, Kent, Cubic, Logistic, Gauss, and Circle were selected [62–69] and used to initialize the population of HHO algorithm. The results were analyzed and the optimal one for the HHO algorithm selected as the population initialization method for the improved algorithm. The following were the mathematical formulas of the 10 chaotic mappings:

(1) Sinusoidal chaotic mapping:

$$\mathbf{x}\_{k+1} = P \cdot \sin(2\pi \mathbf{x}\_k) \tag{14}$$

where *P* was the control parameter, here *P* = 2.3, *x*0 = 0.7, Equation (14) was written as

$$\mathbf{x}\_{k+1} = \sin(\pi x\_k) \tag{15}$$

(2) Tent chaotic mapping

$$\mathbf{x}\_{k+1} = \begin{cases} 2\mathbf{x}\_k & \mathbf{x}\_k < 0.5\\ 2(1 - \mathbf{x}\_k) & \mathbf{x}\_k \ge 0.5 \end{cases} \tag{16}$$

(3) Kent chaotic mapping

$$\begin{cases} \mathbf{x}\_{k+1} = \mathbf{x}\_k / \mu & 0 < \mathbf{x}\_k < \mu \\ \mathbf{x}\_{k+1} = (1 - \mathbf{x}\_k) / (1 - \mu) & \mu \lessapprox \mathbf{x}\_k < 1 \end{cases} \tag{17}$$

The control parameter *μ* ∈ (0, <sup>1</sup>), when *μ* = 0.5, the system was Short Period State, *μ* = 0.5 was not taken here. When using the chaotic mapping, the initial value *x*0 had to not be the same as the system parameters *μ*, otherwise the system evolved into a periodic system. Here, we took *μ* = 0.4.

(4) Cubic chaotic mapping

The standard Cubic chaotic mapping function was expressed as

$$\mathbf{x}\_{k+1} = b\mathbf{x}\_k^3 - c\mathbf{x}\_k \tag{18}$$

where *b* and *c* were the influence factors of chaotic mapping. The range of Cubic chaotic mapping was different for different values to *b* and *c*. When *c* = 3, the sequence generated by Cubic mapping was chaotic. Also when *b* = 1, *xn* ∈ (−2, <sup>2</sup>); when *b* = 4, *xn* ∈ (−1, <sup>1</sup>). Here, we took *b* = 4 and *c* = 3.

(5) Logistic chaotic mapping

$$\mathbf{x}\_{k+1} = P\mathbf{x}\_k(1-\mathbf{x}\_k) \tag{19}$$

when *P* = 4, the generation number of Logistic chaotic mapping was between (0, 1).

(6) Gauss chaotic mapping

$$\begin{array}{ll} \mathbf{x}\_{k+1} = \begin{cases} 0 & \mathbf{x}\_k = 0\\ \frac{1}{\mathbf{x}\_k \bmod (1)} & \text{otherwise} \end{cases} \\\ \frac{1}{\mathbf{x}\_k \bmod (1)} = \frac{1}{\mathbf{x}\_k} - \left| \frac{1}{\mathbf{x}\_k} \right| \end{array} \tag{20}$$

(7) Circle chaotic mapping

$$\mathbf{x}\_{k+1} = \text{mod}\left(\mathbf{x}\_k + 0.2 - \left(\frac{0.5}{2\pi}\right)\sin(2\pi\mathbf{x}\_k), 1\right) \tag{21}$$

The three steps to initialize the population of the HHO using the seven chaotic mappings were:

**Step 1:** Randomly generate *M* Harris hawks in D-dimensional space, i.e., *Y*= (*y* 1, *y*2, *y*3, ··· ··· , *yn*) *yi* ∈ (−1, 1)*i* = 1, 2, ··· , *n*.

**Step 2:** Iterate each dimension of each Harris hawk *M* times, resulting in *M* Harris hawks. **Step 3:** After all Harris hawk iterations were completed, chaotic mapping (21) was applied to the solution space.

$$\mathbf{x}\_{id} = l b\_d + (\mathbf{1} + y\_{id}) \times \frac{u b\_d - l b\_d}{2} \tag{22}$$

where *ub* was the upper bound of the exploration space, *lb* the lower bound of the exploration space; the d-dimensional coordinates of the *i*-th Harris hawk were represented by *yid*, which was generated using Equations (14)–(21); the coordinates of the *i*-th Harris hawk in the d-dimensional of the exploration space were *xid*, which was generated using Equation (22).

Here, we first proposed the HHO algorithm based on seven different chaotic initialization strategies, respectively, chaotic initialization Harris hawks optimization (CIHHO) algorithm. Obviously, the implementation of CIHHO is basically the same as that of HHO, except that the initialization in Step 2 generates *m* individual Harris hawks using Equations (14)–(21), and then maps the positions of these *m* Harris hawks to the search space of the population using Equation (22).
