**2. The Proposed Map**

In [9], the following modified logistic map was proposed:

$$\mathbf{x}\_{i} = 2\beta - \frac{\mathbf{x}\_{i-1}^{2}}{\beta} \tag{1}$$

The behavior of one-dimensional map Equation (1) depends on a single parameter *β*. This map exhibits constant chaos for all values of its parameter, with a full mapping of the state values on *xi* ∈ [−2*β*; 2*β*] provided that *x*<sup>0</sup> ∈ [−2*β*; 2*β*]. Its symmetric bifurcation diagram is shown in Figure 1 and the diagram of its Lyapunov exponent in Figure 2.

**Figure 1.** Bifurcation diagram of Equation (1), with respect to parameter *β*.

**Figure 2.** Diagram of the Lyapunov exponent of Equation (1), with respect to parameter *β*.

Here, a modified version of Equation (1) is proposed, given by

$$\mathbf{x}\_{i} \quad = \quad p\_{i} \left( 2\beta - \frac{\mathbf{x}\_{i-1}^{2}}{\beta} \right) \tag{2}$$

where *pi* = *<sup>r</sup>* · mod (*xi*−1, 1) · (<sup>1</sup> − mod (*xi*−1, 1)). With the above modification, the values of the chaotic map Equation (1) are multiplied by the value *pi* ∈ [0; 1] which is actually the classic logistic map with bifurcation parameter *<sup>r</sup>*, computed using the decimal part of *xi*−1, mod (*xi*−1, 1) ∈ [0; 1] instead of *xi*−1. The mod operator is used here to take the decimal part of *xi*−1, so that *pi* is bounded on the interval [0; 1].

The bifurcation diagram of map Equation (2) with respect to parameter *β* and *r* = 4 is shown in Figure 3. The initial condition in each iteration is chosen as *x*<sup>0</sup> = 0.1. From Figure 3, it can be seen that the system exhibits a similar but more complex behavior compared to Label (1), with small periodic windows appearing. This behavior can be seen more clearly in the zoomed subures. It is observed that the system exhibits crisis phenomena, where it exists abruptly from chaos and reenters it following a period doubling route. What is also interesting is that there are small windows where the phenomenon of antimonotonicity appears. This is when the system enters chaos by following a period doubling route, and then exists from chaos by following a reverse period halving route. This is observed in the subfigures around the value of *β* = 1.02. The chaotic oscillation mode is verified by the diagram of the Lyapunov exponent shown in Figure 4. In addition, Figure 5 shows a full plot for the Lyapunov exponent up to *β* = 150. From this figure, it can be seen that the Lyapunov exponent slowly increases to reach a value higher than 5, while there are also very small periodic windows appearing.

Similar phenomena can be observed for different values of the parameter *r*. For example, the bifurcation diagram and the curve of the Lyapunov exponent with respect to *β* for *r* = 3.8 can be seen in Figures 6 and 7. Again, antimonotonicity appears around the value of *β* = 1.1. The system also exits abruptly from chaos and re-enters it through a period doubling route.

In addition to the rich dynamical behavior with respect to parameter *β*, the proposed map also exhibits chaotic oscillations with respect to parameter *r*, as seen in Figures 8 and 9 where *β* = 10. The system here exhibits crisis phenomena again.

Moreover, as can be seen from Figures 4, 5, 7 and 9, it is important to note that the system can achieve a Lyapunov exponent value that is higher than that of the system in [9] and also the classic logistic map, which both achieve the higher value at around 0.7.

To study the existence of coexisting attractors in the system, its continuation diagram is plotted. The continuation diagram is similar to the bifurcation diagram, with the difference that, in each iteration, the initial value of the chaotic map is taken to be equal to the final value of its previous simulation. The continuation diagram can thus be computed as the bifurcation parameter increases or decreases. Figure 10 shows the bifurcation diagram (black, *x*<sup>0</sup> = 0.1.) of the map with respect to *β* with *r* = 4, overlapping with its forward (red) and backward (green) continuation diagrams. This plot reveals coexisting attractors for the system around the value of *β* = 1.05. This means that, depending on the initial condition of the system, its steady-state behavior may converge to different attracting regions.

**Figure 3.** Bifurcation diagram of Equation (2), with respect to parameter *β*, for *r* = 4.

**Figure 4.** Diagram of the Lyapunov exponent Equation (2), with respect to parameter *β*, for *r* = 4.

**Figure 5.** Wider diagram of the Lyapunov exponent Equation (2), with respect to parameter *β*, for *r* = 4.

**Figure 6.** Bifurcation diagram of Equation (2), with respect to parameter *β*, for *r* = 3.8.

**Figure 7.** Diagram of the Lyapunov exponent Equation (2), with respect to parameter *β*, for *r* = 3.8.

**Figure 8.** Bifurcation diagram of Equation (2), with respect to parameter *r*, for *β* = 10.

**Figure 9.** Diagram of the Lyapunov exponent Equation (2), with respect to parameter *r*, for *β* = 10.

**Figure 10.** Bifurcation diagram (black), forward continuation diagram (red), and backward continuation diagram (green) for Equation (2), with respect to parameter *β*, for *r* = 4.
