*2.2. Stability Analysis*

For discrete-time systems, the fixed point of a function form a graphical point of view is an element in the domain that maps to itself by the function. For instance, *P* is a fixed point of the function *F*(*x*) only when *Fn*(*P*) = *P*. To simplify the calculation of the fixed points of 2D-ICSM, we reduce its dimension to become 1D as follows,

$$y^{(v)} = (a+2)\sin\left(\beta y^{(v)}\right) + \sin\left(\frac{\beta}{y^{(v)}}\right),\tag{3}$$

The fixed points of the 2D-ICSM are calculated for two different sets of system parameters. For each set of parameters, we obtain the fixed points of the variable *y* by Equation (3), and subsequently, the corresponding points of the variable *x* can be easily obtained by the first equation of the system (2). Figure 2 illustrates how the fixed points of the variable *y* can be obtained using the graphical method. From this figure, one can notice that the number of fixed points is increased as the values of the amplitude and internal frequency parameters increase. Now, let us collect some fixed points from each

set of system parameters to investigate their stability. First, we have extracted the following fixed point from the first set of the system parameters,

$$\left\{ P\_1 = \left( \mathfrak{x}^{(1)}, \mathfrak{y}^{(1)} \right) = (0.6687, 2.4092) \right\}$$

Second, we have extracted three different fixed points from the second set as follows,

$$\begin{cases} P\_1 = \left(\mathbf{x}^{(1)}, \mathbf{y}^{(1)}\right) = (0.9168, 4.1324), \\ P\_2 = \left(\mathbf{x}^{(2)}, \mathbf{y}^{(2)}\right) = (0.7607, 3.5738), \\ P\_3 = \left(\mathbf{x}^{(3)}, \mathbf{y}^{(3)}\right) = (0.1334, 1.5039). \end{cases}$$

**Figure 2.** Fixed points of the 2D-ICSM: (**a**) for the parameters *α* = *β* = 1; (**b**) for the parameters *α* = *β* = 2.

The stability of the above-fixed points can be determined by obtaining the Jacobian matrix, which is given by

$$J = \begin{pmatrix} \frac{\partial f\_1}{\partial x} & \frac{\partial f\_1}{\partial y} \\\\ \frac{\partial f\_2}{\partial x} & \frac{\partial f\_2}{\partial y} \end{pmatrix}.$$

Using the above matrix, the 2D-ICSM is Linearized at any arbitrary fixed point *Pi* = (*x*∗, *y*∗) as follows,

$$J\_{P\_i} = \begin{pmatrix} 0 & \beta \cos(\beta y) \\ 2+a & -\frac{\beta}{y^2} \cos\left(\frac{\beta}{y}\right) \end{pmatrix} \cdot \mathbf{1}$$

Thus, the eigenvalues are obtained by solving the following equation,

$$
\lambda^2 + \left(\frac{\beta}{y^2}\cos\left(\frac{\beta}{y}\right)\right)\lambda - (2+a)\beta\cos(\beta y) = 0.
$$

It is well-known that the stability of fixed points is dependent on the eigenvalues. When an eigenvalue is within the interval [−1, 1], then the fixed point exhibits a stable state. Otherwise, it shows an unstable state. Moreover, the stability of the obtained fixed points is as illustrated in Table 1. All the selected fixed points of the 2D-ICSM are unstable.


**Table 1.** The fixed points of the 2D infinite-collapse-Sine model (2D-ICSM) and their stability analysis.

#### **3. Dynamical Behaviors**

This section investigates the dynamical behaviors of 2D-ICSM through the bifurcation diagram, Lyapunov Exponents, and phase space.
