## 2 The Proposed Function Composition
### Behavior Analysis  
This section is aimed at showing the behavior analysis for the proposed chaotic maps. To such purpose, we identify the conditions for which the chaotic maps can generate periodic sequences, as well as those conditions for which they can generate aperiodic sequences. In order to offer this analysis of behavior (periodic or aperiodic), in a similar way to Palacios-Luengas et al. [14] and Pichardo-Mendez et al. [67], we identify the chaos annulling conditions in the proposed chaotic maps, and we calculate their bifurcation diagrams and Lyapunov exponents. Firstly, to identify the chaos annulling conditions in the proposed chaotic maps, we must find their fixed points and their periodic orbit of order \(m>2\). For this intention, we assume that \(x^{(1)}\) is a fixed point of the chaotic system \(\xi(\cdot)\) when \(\xi\left(\mu,\alpha,x^{(1)}\right)=x^{(1)}\), \(x^{*}\) is a preimage point when \(x^{(1)}=\xi(\mu,\alpha,x^{*})\), and \(x^{(m)}\) is a condition that produces an periodic orbit of order \(m>2\) when \(\xi^{m}\left(\mu,\alpha,x^{(m)}\right)=x^{(m)}\), considering that \(\xi^{m}(\mu,\alpha,\hat{x})\) is \(m_{th}\) iteration of \(\xi(\cdot)\) when \(\hat{x}\) is its initial condition. As an example for the proposed chaotic maps, Table 1 shows the fixed points and some periodic orbits of order \(m=2,3\), and \(4\). Note in Table 1 that the examples of conditions producing periodic orbit of order \(m=2,3\) and \(4\) have be included by using only ten digits after the dot, but they are number with more significant digits. Additionally, note that Table 2 shows the preimage points. the bifurcation diagrams for the FC. Remember that, a bifurcation diagram illustrates the changes that occurred to the number sequences produced by a chaotic system considering different values of its control parameters. In Figures 2a and 3a, \(\mu\in\left(0,\,3.0\right)\) and \(\alpha=3.0\), meanwhile in Figures 2b and 3b \(\mu=0.5\) and \(\alpha\in\left(0,\,3.0\right)\), in Figures 2c and 3c \(\mu=1.5\) and \(\alpha\in\left(0,\,3.0\right)\), and in Figures 2d and 3d \(\mu=2.5\) and \(\alpha\in\left(0,\,3.0\right)\). In all cases, the initial conditions were randomly selected for each value of control parameter used when the chaotic system was being iterated. According to Table 1, Figure 2 shows the fixed points (red lines) and some periodic orbits for the STM. For all cases, **A** corresponds to \(x\)=\(\alpha\), and **B** to \(x=\frac{\alpha^{2}}{\left(2\alpha-\mu\right)}\). But in Figure 2b, **C** corresponds to \(x=\frac{2\alpha^{2}}{\left(8\alpha^{3}-4\alpha^{2}+4\alpha-1\right)}\), and **D** to \(x=\frac{2\alpha^{2}}{\left(16\alpha^{4}+2\alpha-1\right)}\); in Figure 2c, **C** corresponds to \(x=\frac{8\alpha^{4}}{\left(8\alpha^{4}+18\alpha-27\right)}\), and **D** to \(x=\frac{54\alpha^{2}}{\left(16\alpha^{4}+54\alpha-81\right)}\); and in Figure 2d, **C** corresponds to \(x=\frac{50\alpha^{2}}{\left(8\alpha^{3}-20\alpha^{2}+100\alpha-125\right)}\), and **D** to \(x=\frac{5.25968\times 10^{14}\alpha^{2}}{\left(3.36619\times 10^{15}\alpha^{4}+5.25968 \times 10^{14}\alpha^{4}-1.34192\times 10^{15}\right)}\). On the other hand, Figure 3 shows the fixed points (red lines) to \(x\)=\(\alpha\) and, additionally, Figure 3c indicates a stability island. The fixed points periodic orbits and stability islands must be identified and avoided when the chaotic system is applied in cryptosystems. Additionally, note that, the bifurcation diagrams in Figures 2a and 3a completely cover the plane \(\mu\)_vs_\(\tau(\mu,\alpha,x)\); but, the bifurcation diagram for the FC does not exhibit the annulling chaos conditions that are present on the STM, which are given by \(x_{0}=\mu\) or \(x_{0}=\alpha\), and \(x_{0}=0\). Figure 2: Bifurcation diagram of the STM: (**a**) \(\mu\in\left(0,\,3.0\right)\) and \(\alpha=3.0\), (**b**) \(\alpha\in\left(0,\,3.0\right)\) and \(\mu=0.5\), (**c**) \(\alpha\in\left(0,\,3.0\right)\) and \(\mu=1.5\), and (**d**) \(\alpha\in\left(0,\,3.0\right)\) and \(\mu=2.5\).