*2.1. The Structure of Chaotic Maps*

First, this section reviews three chaotic maps including tent map, sine map, and logistic map. Based on these three chaotic maps, the TDSCL map is generated. The tent map is defined mathematically as [22]:

$$\mathbf{x}\_{n+1} = T\_{\lambda}(\mathbf{x}\_n) = \begin{cases} 2\lambda \mathbf{x}\_n & \text{for } \mathbf{x}\_n < 0.5 \\\ 2\lambda (1 - \mathbf{x}\_n) & \text{for } \mathbf{x}\_n \ge 0.5 \end{cases} \tag{1}$$

where *λ* is the control parameter with the range of [0, 1].

The structure of the sine map is defined as [23]:

$$\mathbf{x}\_{n+1} = \mathbf{S}\_{\mathfrak{a}}(\mathbf{x}\_{\mathfrak{n}}) = a \sin(\pi \mathbf{x}\_{\mathfrak{n}}) \tag{2}$$

where *α* is the control parameter with a range of [0, 1], and the map is chaotic with *α* ∈ (0.87, 1) . For all *n* ≥ 1, *xn* is bounded within [0, 1]. The diagrams of bifurcation are shown in Figure 1b.

The logistic map is a simple 1D chaotic map. As a discrete chaotic map, Figure 1c shows its bifurcation, with outputs in the range of [0, 1] and an initial input value in [0, 1]. The structure of the logistic map is defined by [24]:

$$\mathbf{x}\_{n+1} = L\_{\mu}(\mathbf{x}\_{n}) = 4\mu \mathbf{x}\_{\hbar} (1 - \mathbf{x}\_{\hbar}) \tag{3}$$

where *μ* is the control parameter in the range of [0, 1].

**Figure 1.** The bifurcation diagram for the (**a**) tent map, (**b**) sine map, and (**c**) logistic map.

The above three chaotic maps all have flaws, producing no chaotic behavior in some ranges of parameters. Specifically, these three maps only show chaotic characteristics at the rightmost part of the parameter variation range, and the chaotic interval may be discontinuous. To overcome these flaws, we designed a novel chaotic map structure, which is shown in Figure 2. As shown in Figure 2,

*T*(*x*) represents the tent map with a delay item input, and the sine map is indicated by *S*(*x*). Then, the outputs of *T*(*x*) and *S*(*x*) are added as the input of *f*(*x*). The function *f*(*x*) is taken as *e<sup>x</sup>* in this paper, and cascaded with *L*(*x*), thereby obtaining the output result of the chaotic map.

$$\mathbf{x}\_{n+1} = \mu f \circ F(\mathbf{x}\_n)(1 - f \circ F(\mathbf{x}\_n)) \, mod \, \mathbf{1} \tag{4}$$

$$F(\mathbf{x}\_n) = \begin{cases} 2\mathbf{x}\_{n-1} + \sin(\pi \mathbf{x}\_n) & \mathbf{x}\_n < 0.5\\ 2(1 - \mathbf{x}\_{n-1}) + \sin(\pi \mathbf{x}\_n) & \mathbf{x}\_n \ge 0.5 \end{cases} \tag{5}$$

**Figure 2.** The structure of the tent delay-sine cascade with logistic map (TDSCL).

Here, the control parameters for the tent map and the sine map are set to 1, and the parameter *μ* for the logistic map is used as the control parameter for this new map. Equations (4) and (5) show the mathematical formulae. The circle symbol in Equation (4) represents the composition of two functions. Compared to the 1D delay and linearly coupled logistic chaotic map (DLCL) [15] and a two-dimensional logistic-modulated sine-coupling logistic chaotic map (LSMCL) [1], the structure of TDSCL produces better chaotic performance. In the following section, we use the trajectory, Lyapunov exponent, and permutation entropy (PE) to analyze the characteristics of chaotic maps.

## *2.2. Chaotic Performance of TDSCL*
