# Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems

## Abstract

**:**

## 1. Introduction

## 2. Exponential Polynomial Family of DDS

**Definition**

**1.**

**Definition**

**2.**

## 3. Analysis of the Family

#### 3.1. Perturbed Linear Case, $\beta =1$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Remark**

**1.**

#### 3.2. Basic Exponential Polynomial Case, $\beta =0.$

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**3.**

**Remark**

**2.**

#### 3.3. Mixed Case $\beta \in (0,1)$

## 4. Transient Behavior for the Case $\mathit{\beta}=\mathbf{0}$

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphs of the iteration function $f(\alpha ,\beta ,x)$ with $\alpha =4$ and with different values of $\beta .$

**Figure 2.**Basin of attraction of the fixed point $x=2$ for $\alpha \in (0,7)$ and initial conditions ${x}_{0}\in [1.5,2.2]$. Colors denotes the number of iterations within a tolerance of ${10}^{-3}$.

**Figure 3.**Bifurcation diagram of System (2) with initial conditions (

**a**) 0.05, (

**b**) 0.19, (

**c**) 1.7. Horizontal axis is $\alpha $ and vertical axis is ${x}_{n}$. $n>3000$ for (

**a**) and $n>8000$ for (

**b**,

**c**).

**Figure 4.**Number of iterations required to obtain convergence to the fixed point given a tolerance of ${10}^{-3}$ for $\alpha \in (0,1)$ and initial condition ${x}_{0}=0.5$.

**Figure 5.**(

**a**) Fixed points for System (2) when $\beta =0.$ (

**b**) Bifurcation diagram with initial condition ${x}_{0}=0.3$ and 1000 initial iterations discarded.

**Figure 6.**(

**a**) Graph of $b\left(\beta \right)$. (

**b**) Bifurcation diagram for $\beta =0.3$. (

**c**) Bifurcation diagram for $\beta =0.6$.

**Figure 7.**Bifurcation diagrams for $\beta =0$ and initial condition ${x}_{0}=0.3.$ Horizontal axis is $\alpha $ and vertical axis is ${x}_{n}$. (

**a**) First 1000 iterations discarded. (

**b**) First 5000 iterations discarded. (

**c**) Both diagrams (

**a**) and (

**b**).

**Figure 8.**Bifurcation diagrams for $\beta =0$ and initial condition ${x}_{0}=0.3.$ Horizontal axis is $\alpha $ and vertical axis is ${x}_{n}$.

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**MDPI and ACS Style**

Solis, F. Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems. *Math. Comput. Appl.* **2019**, *24*, 13.
https://doi.org/10.3390/mca24010013

**AMA Style**

Solis F. Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems. *Mathematical and Computational Applications*. 2019; 24(1):13.
https://doi.org/10.3390/mca24010013

**Chicago/Turabian Style**

Solis, Francisco. 2019. "Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems" *Mathematical and Computational Applications* 24, no. 1: 13.
https://doi.org/10.3390/mca24010013