*2.1. Definition of 2D-ICSM*

Among existing 1D discrete-time dynamical systems, the infinite collapse model is considered as one of the best maps that show robust chaotic performance [38]. Mathematically, its dynamical equation is given by

$$x\_{n+1} = \sin\left(\frac{\beta}{x\_n}\right),\tag{1}$$

where *β* is the control parameter and *x* is the state variable. The dynamical behavior of this map can be illustrated by depicting its bifurcation diagram and trajectory in the phase plane, as shown in Figure 1. It can be seen that the numerical solution of this map is in the range of [−1, 1]. Besides that, the bifurcation diagram of this map shows that the chaotic attractor appears in limited regions of its parameter. Meanwhile, several non-chaotic regions can be observed as the parameter *α* increasing. Furthermore, Figure 1b demonstrates that its trajectory only occupies a small space in the phase plane. Such behaviors are widely observed in the existing 1D chaotic maps such as Logistic, Tent, and Sine maps.

**Figure 1.** Dynamical behavior of the infinite collapse map (1) with *x*<sup>0</sup> = 0.5: (**a**) bifurcation diagram for *β* ∈ [0, 8] and (**b**) chaotic attractor for *β* = 2.

To tackle the aforementioned issues, we propose a new 2D chaotic map, which consists of four terms with two control parameters. Mathematically, it is defined as follows,

$$\begin{cases} \mathbf{x}\_{n+1} = \sin\left(\beta y\_n\right), \\ y\_{n+1} = (\alpha + 2)\mathbf{x}\_n + \sin\left(\frac{\beta}{y\_n}\right), \end{cases} \tag{2}$$

where *α* is the amplitude parameter and *β* is the internal frequency parameter. It can be seen that the proposed 2D infinite-collapse-Sine model (2D-ICSM) is mainly designed by using three components including a linear variable, 1D Sine map, and 1D infinite collapse map. The linear state variable *xn* is used to modulate the output of a 1D infinite collapse map. Therefore, it can enhance the chaotic behavior of the state variable *yn*+1. Meanwhile, the 1D Sine map is employed to boost randomness to the state variable *xn*+1.
