*3.1. Bifurcation Diagram and Lyapunov Exponents*

Typically, the bifurcation diagram and Lyapunov Exponents are used to determined the non-chaotic and chaotic regions of a dynamical system when one of its parameters varies. Furthermore, the Lyapunov exponent is used to evaluate the chaotic properties of a dynamical system. In other words, it could recognize the chaotic and hyperchaotic behaviors of the system. A system is recognized as chaotic when there is one positive Lyapunov Exponent value for each parameter value, whereas the hyperchaotic system has more than one positive Lyapunov Exponent value. The hyperchaotic system exhibits a higher level of randomness, and the generated sequences by the hyperchaotic system show extreme unpredictability.

To investigate the dynamics of 2D-ICSM, we depict its bifurcation diagram and Lyapunov Exponents with the initial values (0.5, 0.5) and for the parameters 0 ≤ *α* ≤ 8 and *β* = 12, as shown in Figure 3. It can be seen that 2D-ICSM is hyperchaotic among the whole parameter range, which indicates that its sequences are extremely hard to be predicted.

**Figure 3.** Dynamics of the 2D-ICSM with the initial values (0.5, 0.5) and for the parameter *β* = 12: (**a**) bifurcation diagram; (**b**) Lyapunov Exponents.

## *3.2. Hyperchaotic Attractor*

The set of numerical values, which is generated by a chaotic/hyperchaotic map with specific sets of initial values and control parameters, is called chaotic/hyperchaotic attractor. For a 2D map, its attractor can be described by a group of points that occupies a particular region in the phase space. A chaotic/hyperchaotic model has better performance when its attractor is geometrically complicated and occupies a larger range in the phase space. To illustrate the hyperchaotic range of the 2D-ICSM, Figure 4f depicts its attractor in the 2D phase space with the parameters *α* = 6 and *β* = 12. Besides that, this figure plots the attractors of several existing chaotic and hyperchaotic models to demonstrate the complicated behavior of the 2D-ICSM. It can be observed that the hyperchaotic attractor of 2D-ICSM fully occupies a 2D phase space ranging *x* ∈ [−1, 1] and *y* ∈ [−9, 9]. This means that 2D-ICSM

can generate more unpredictable hyperchaotic sequences and it has a better competitive ergodicity property than existing models.

**Figure 4.** Chaotic and hyperchaotic attractors of different 2D maps: (**a**) 2D-SLMM [12]; (**b**) 2D-SIMM [13]; (**c**) 2D Ushiki map [39]; (**d**) 2D-LASM [14]; (**e**) 2D-LICM [15]; (**f**) the 2D-ICSM.
