## 2 The Proposed Function Composition
### Behavior Analysis
Red and black lines show the fixed points, and green and magenta lines show the conditions for periodic orbits. Now, in order to analyze the stability island of order 3 in the FC, we have generated Figure 4, which shows in detail the window for the stability island of order 3 identified at Figure 3. This window allows us to identify the auto-similarity property of the FC and, according to Sharkovski's Theorem [68; 69], it ensures that the FC has an infinite number of stability islands. Therefore, in the FC the stability islands emerge according to Sharkovski sequence \(2^{n}\times k\), with \(k=3\), \(5\), \(7\), \(9\),... for \(n\)= 1, 2, 3, 4,... [68; 69], and they are inherited from the sine chaotic map. All stability islands must be considered and avoided. It is worth noting that the period doubling phenomenon observed in Figure 3c, boxed in red, appears again in Figure 4a. Figure 4b shows a zoom at region boxed in red at Figure 4a. Figure 4c,d have been included in order to show that the auto-similarity property, the doubling period phenomenon and the stability island of order 3 appear again. This is sufficiently clear evidence that the FC has an infinite number of stability islands, which emerge according to the Sharkovski sequence. We highlight that, if \(\mu\) is considered the main control parameter, both chaotic systems exhibit a chaotic behavior when \(\mu\in(0,\alpha)\); meanwhile, if \(\alpha\) is considered the main control parameter, they exhibit a chaotic behavior when \(\alpha>\mu\), and this behavior is limited by the function \(\tau=\alpha\). Additionally, the statistical distribution of the STM is closer to the uniform distribution than the statistical distribution of the FC, which is denser at ends. But, the FC does not have fixed points and stability islands at parameter domains of chaotic behavior when \(\mu<\alpha\). In a similar way to Palacios-Luengas et al. [14] and Pichardo-Mendez et al. [67], the statistical distribution of the sequences produced by both chaotic maps can be estimated through the stationary statistical distribution by using the Birkhoff's Ergodicity Theorem [70], and considering that the evolution of a set of initial condition must be studied when the chaotic system is applied to it. Now, as an example, in Figure 5 we show the trajectory diagrams of both chaotic systems when \(\alpha=3.0\) considering 10,000 iterations. For the STM, in Figure 5a, we use \(\mu=0.9\) and \(x_{0}=1.9\), and, in Figure 5b, we show that a short trajectory reaches a fixed point when \(\mu=1.5\) and \(x_{0}=0.0625\). On the other hand, for the FC, in Figure 5c, we show that a chaotic trajectory occurs when \(\mu=0.9\) and \(x_{0}=1.9\), and, in Figure 5d, we show that a short trajectory reaches a fixed point when \(\mu=1.5\) and \(x_{0}=1.460321868288294\). The next step, in the behavior analysis of a chaotic system is to define whether the fixed points or the periodic orbits are stable (attractor) or unstable (repeller) points or orbits. Analyzing the stability of the fixed points, let \(|\eta_{n}|=|\xi^{n}(\mu,\alpha,x_{0}+\eta_{0})-\xi^{n}(\mu,\alpha,x_{0})|\) be the relative difference between the values of the position \(n\) in the number sequences produced by the chaotic system \(\xi(\cdot)\), where \(\xi(\cdot)\)=\(\sigma(\cdot)\) for the STM and \(\xi(\cdot)\)=\(\tau(\cdot)\) for the FC. Using \(x_{0}+\eta_{0}\) and \(x_{0}\) as initial conditions to produce two number sequences, let \(\eta_{0}\) be some number arbitrarily small, and \(\mu\) and \(\alpha\) the control parameters of the chaotic systems. If \(|\eta_{n+1}|<|\eta_{n}|\), then the selected control parameter will cause the chaotic map to converge \(\forall\)\(n\), causing the produced number sequence to fall at some fixed point, which will be an attractor fixed point. Conversely, if \(|\eta_{n+1}|>|\eta_{n}|\), then the selected control parameter will cause the chaotic map to diverge \(\forall\)\(n\), causing the produced number sequence to move away from the fixed point, which will be a repeller fixed point.