## 2 The Proposed Function Composition
### Behavior Analysis
This derivative criterion for  
Figure 4: Bifurcation diagram of the FC when \(\mu=1.5\). (**a**) \(\alpha\in(1.405,1.425)\), it corresponds to a close-up on Figure 3c. (**b**) \(\alpha\in(1.408,1.422)\), it corresponds to a close-up on (**a**). (**c**) \(\alpha\in(1.402,1.4205)\), it corresponds to a close-up on (**b**). (**d**) \(\alpha\in(1.402,1.4205)\), it corresponds to a close-up on (**c**). repelling and attracting fixed points can be generalized to periodic orbits. For this purpose, we recommend reviewing Ref. [68]. Now, as an example of this concepts, we analyzed the cases when \(x_{0}\) is a fixed point, in order to estimate the trap conditions. Thus, let \(x^{*}\) be a fixed point, and considering that \(\tau^{n}(\mu,\alpha,x^{*})=x^{*}\)\(\forall\)\(n\), the relative difference \(\eta_{n+1}\) can be written as,  
\[ \begin{split}|\eta_{n+1}|=&|\xi^{n}(\mu,\alpha,x^{* }+\eta_{0})-\xi^{n}(\mu,\alpha,x^{*})|,\\ =&|\xi^{n}(\mu,\alpha,x^{*}+\eta_{0})-x^{*}|.\end{split}  \tag{5}\]  
Figure 5: Trajectory diagrams. (**a**) STM when \(\alpha=3.0\), \(\mu=0.9\), and \(x_{0}=1.9\) (chaotic trajectory), (**b**) STM when \(\alpha=3.0\), \(\mu=1.5\), and \(x_{0}=0.0626\) (short trajectory reaches the fixed point \(x^{*}=2.0\)), (**c**) FC when \(\alpha=3.0\), \(\mu=0.9\), and \(x_{0}=1.9\) (chaotic trajectory), and (**d**) FC when \(\alpha=3.0\), \(\mu=1.5\), and \(x_{0}=1.460321868288294\) (short trajectory reaches the fixed point \(x^{*}=3.0\)). By Taylor's series,  
\[ \begin{split}|\eta_{n+1}|=&\left|\xi^{n}(\mu,\alpha,x^{*} )-\eta_{n}\frac{d\xi(\mu,\alpha,x)}{dx}\right|_{x^{*}}-x^{*}\right|,\\ =&\left|\eta_{n}\frac{d\xi(\mu,\alpha,x)}{dx}\right|_ {x^{*}}\Bigg{|},\\ =&|\eta_{n}\xi^{\prime}(\mu,\alpha,x^{*})|.\end{split}  \tag{6}\]  
Therefore, \(\eta_{n+1}<\eta_{n}\) occurs when \(\xi^{\prime}(\mu,\alpha,x^{*})<1\) and \(x^{*}\) is a fixed attractor point, and \(\eta_{n+1}>\eta_{n}\) occurs when \(\xi^{\prime}(\mu,\alpha,x^{*})>1\) and \(x^{*}\) is a repeller fixed point. Then, considering Table 1 for \(\sigma(\mu,\alpha,\cdot)\), and according to Equation (7), \(x^{*}=0\) is a repeller point because \(\sigma^{\prime}(\mu,\alpha,x)>1\) and \(0<\mu<\alpha\). We can verify this condition using Figure 5b, in which \(\mu=1.5\), \(\alpha=3.0\), and \(x=0.0625\), and the trajectory (number sequence) reaches the fixed point \(x^{*}=2.0\). In a similar way, according to Equation (7), \(x^{*}=\frac{\alpha^{2}}{2\alpha-\mu}\) is a repeller point if \(\sigma^{\prime}(\mu,\alpha,x)>1\), and this condition occurs when \(1.0<\alpha<2.0\) and \(2\alpha-\alpha^{2}<\mu\), or when \(2.0<\alpha\). But it is an attractor point when \(0.0<\alpha<1.0\), or when \(1.0<\alpha<2.0\) and \(0.0<\mu<2\alpha-\alpha^{2}\). \[\sigma^{\prime}(\mu,\alpha,x)= \begin{cases}\frac{\alpha}{\mu}&0<x\leq\mu\\ \frac{\alpha}{\mu-\alpha}&\mu<x<\alpha^{\prime}\end{cases}  \tag{7}\]  
On the other hand, considering Table 1 for \(\tau(\mu,\alpha,\cdot)\), and according to Equation (8), \(x^{*}=0.0\) is a repeller point when \(0<\alpha\leq\mu\) or when \(\alpha<\mu+\pi acos\left(\frac{\pi(\mu-2\alpha)}{\alpha-\mu}\right)\), and it will be an attractor point when \(\mu+\pi acos\left(\frac{\pi(\mu-2\alpha)}{\alpha-\mu}\right)<\alpha\). In a similar way, \(x^{*}=\alpha\) will be a repeller point when \(\pi acos\left(\frac{\pi\alpha}{\mu}\right)>\mu\), and it will be an attractor point when \(\pi acos\left(\frac{\pi\alpha}{\mu}\right)<\mu\). Finally, \(x^{*}=\mu\) is a repeller point because \(\frac{\pi\alpha}{\mu}\) is always greater than \(1.0\). \[\tau^{\prime}(\mu,\alpha,x)= \begin{cases}\frac{\pi\alpha}{\mu}\cos\left[\pi \frac{\gamma}{\mu}\right]&0<x\leq\mu\\ \frac{\pi\alpha}{\alpha-\mu}\cos\left[\pi\frac{\gamma-2\alpha+\mu}{\alpha-\mu }\right]&\mu<x<\alpha^{\prime}\end{cases} , \tag{8}\]  
In this way, a chaotic system will have a CAT condition when the sequences it produces reach an attractor fixed point. Therefore, it is very important to know and avoid fixed points in a chaotic system when it intends to be incorporated in cryptosystems.