## 2 The Proposed Function Composition
### Sensitivity Analysis  
A very effective way of determining the chaos annulling traps (CATs) in a chaotic system is by performing a sensitivity analysis. For this analysis, the Lyapunov exponent, \(\lambda\), helps to detect a chaotic behavior in systems because it quantifies the separation rate of infinitesimally close trajectories in its phase space. In a similar way, to analyze the fixed points, \(\lambda\) is calculated considering that \(|\eta_{n}|=\eta_{0}e^{n\lambda}\). Note that if \(\lambda>0\), the two trajectories produced by the chaotic map will diverge when the separation of their initial conditions is arbitrary small. In this case, the map has a chaotic behavior, and in the case of \(\lambda<0\) the map will have a non-chaotic behavior. Although there are other approaches to calculate the Lyapunov exponent such as using unstable periodic orbits [74], we decided to apply the following numerical approximation used by Palacios et al. [14]. \[\lambda\approx\frac{1}{n}\sum_{i=0}^{n-1}ln|\xi^{\prime}(\mu,\alpha,x_{i})|. \tag{9}\]  
Note that, Equation (9) represents the maximum value of the velocity average, in exponential order, with which a first trajectory produced by a chaotic map moves away (or approaches) from other trajectories generated by the same map from an initial condition very close to the one used to produce the first trajectory. From Equation (9), and considering that for the STM, \(\sigma^{\prime}(\mu,\alpha,x_{i})\) is defined by Equation (7), and for the FC, \(\tau^{\prime}(\mu,x_{i})\) is defined by Equation (8), we have calculated \(\lambda\) for both chaotic maps and we showed in Figure 6 their behavior as a function of the control parameters, \(\mu\) and \(\alpha\). Figure 6a shows \(\lambda_{\sigma}\) as a function of \(\mu\in(0,\,\alpha)\) and \(\alpha=3.0\) for the STM. Note that \(\lambda_{\sigma}>0\)\(\forall\)\(\mu\). In a similar way, Figure 6b shows \(\lambda_{\tau}\) as a function of \(\mu\in(0,\,\alpha)\) and \(\alpha=3.0\) for the FC. Also, note that \(\lambda_{\tau}>0\)\(\forall\)\(\mu\). Oppositely, Figure 6c shows \(\lambda_{\sigma}\) as a function of \(\alpha\in(0,\,3.0)\) and \(\mu=0.5,\,1.5\) and \(2.5\) for the STM. Note that \(\lambda_{\sigma}>0\) when \(\alpha>\mu\). In a similar way, Figure 6d shows \(\lambda_{\tau}\) as a function of \(\alpha\in(0,\,3.0)\) and \(\mu=0.5,\,1.5,\) and \(2.5\) for the FC. Also, note that \(\lambda_{\tau}>\)0 when \(\alpha>\mu\). According to results showed in Figure 6d, we must emphasize that the FC exhibits islands of stability only when \(\mu_{critical}<\alpha<\mu\) in the chaotic map; assuming that \(\mu_{critical}\) is the value of \(\mu\) when \(\lambda\) crosses zero for the first time. Thus, if \(\alpha\in(0,\,3.0)\) for the FC, it will always be true that \(\lambda_{\sigma}>0\) when \(\alpha>\mu\) is satisfied avoiding that chaotic map generates chaos annulling conditions. On the other hand, if \(\mu\in(0,\,3.0)\) for the FC, it will always be true that \(\lambda_{\tau}>0\) when \(\mu\in(0,\,\alpha)\). Therefore, when \(\mu\in(0,\,\alpha)\), the FC will produce chaotic sequences. selected small number. Therefore, \(N_{th}\) is the tolerance level of the chaotic map as a function of \(k\), where \(k\) is the smallness of \(\epsilon_{0}\). In summary, then, Figure 8a shows \(N_{th}\) versus \(k\) for the FC (blue lines) and the STM (red lines), considering that \(k\in[1,\,15]\), \(x_{0}=0.1\), \(\mu=1.0\), \(\alpha=2.0\), and \(n=1000\) with \(\delta=1\times 10^{-3}\) (\"\(\blacksquare\)\") and \(\delta=1\times 10^{-5}\) (\"\(\blacktriangle\)\"). Note that the FC has a smaller tolerance level than the STM for changes in the initial conditions, because when using the same \(k\), in both chaotic maps, \(N_{th}\) for the FC is smaller than \(N_{th}\) for the STM. Now, using these concepts, we define the sensitivity level according to Equation (10). In Figure 8b, we show the behavior of \(L(\delta,k)\) for both chaotic maps when \(k\in[1,\,15]\) with \(\delta=1\times 10^{-3}\) and \(\delta=1\times 10^{-5}\).