Analysis of Dynamical Behavior of One-Dimensional Real Maps: An Executable Dynamical Programming Software Approach

Mohammad Sharif Ullah^1,2*, Masuda Akter^1,2, K. M. Ariful Kabir²

¹Department of Mathematics, Feni University, Feni, Bangladesh.
²Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh.

DOI: 10.4236/am.2023.149038                

Abstract

The dynamical behavior of real-world phenomena is implausible graphically due to the complexity of mathematical coding. The present article has mainly focused on some one-dimensional real maps’ dynamical behavior irrespective of using coding. In continuation, linear, quadratic, cubic, higher-order, exponential, logarithmic, and absolute value maps have been used to scrutinize their dynamical behavior, including the characteristics of the orbit of points. Dynamical programming software (DPS.exe) will be proposed as a new technique to ascertain the dynamical behavior of said maps. Thus, a mathematician can automatically determine one-dimensional real maps’ dynamical behavior apart from complicated programming code and analytical solutions.

Keywords

One-Dimensional Map, Cobweb, Orbit Diagram, Fixed Point, the Fate of the Orbit

1. Introduction

Mathematical equations analytically reveal the idea of numerous expectations for modeling all-natural phenomena. In this regard, dynamical systems demonstrate a significant role and have an extensive and substantial aspect expressed by prominent mathematicians [1] - [12] .

One-dimensional maps play a crucial role [13] [14] [15] in predicting the natural behavior and physical object’s fate more precisely. The readers advised reading [16] [17] to know the periodic and aperiodic actions in discrete one-dimensional dynamical systems and the history of one-dimensional dynamics. Many studies [18] [19] [20] address real maps and dynamical behaviors with different approaches. Clark et al. prove the complex box bounds for real maps [21] . Iwanaga and Namatame signified evacuation decision-making contagion on a real map [22] . Jia et al. proposed a mobility model based on a real map for VANETs to overwhelm the existing model’s disadvantages [23] . Joshi and Blackmore effectively modeled the discrete evolution of space, biological, and ecological sciences by exponentially decaying discrete dynamical systems [24] .

Furthermore, many studies [25] [26] [27] [28] investigated one-dimensional map characteristics under different conditions. Sushko et al. discussed some basic concepts and definitions of non-smooth one-dimensional maps [29] . Some studies [30] - [35] introduced new techniques to discover dynamical map features. Medrano and Solis extended and improved the existing characterization of general quadratic actual polynomial maps dynamics with coefficients [36] . Bai et al. [37] analyze the invariant solutions of Coupled Burgers’ equations utilizing one-dimensional optimum systems. The ground-state energy and entropy for a one-dimensional Heisenberg chain with alternating D-terms are investigated by Xiang et al. [38] .

Moreover, analyzing dynamical behaviors, such as fixed-point, iteration, orbit under specific values, and the orbit’s fate, is challenging due to the complicated mathematical calculation and programming codes [39] [40] . Therefore, in the present study, one-dimensional real map-based techniques are proposed to determine their dynamical behavior without complicated programming, compressing a mathematician or physicist’s effort.

The progression of the current research work is as follows. The formulation is thoroughly described in Section 2. Section 3 offers a numerical and graphical discussion of the maps mentioned earlier. On top of that, we provide a detailed comparison between numerical, visual, and DPS.exe analysis. The final words are given in Section 4.

2. Methodology

In any research, one of the best unspoken tools is arriving at reliant elucidations to the problems through systematic assortment and analysis. Firstly, the dynamical behavior of one-dimensional stated maps is discussed using different coding software [41] - [45] . Then, an executable FORTRAN coding system is used in the background of the newly proposed software. In this regard, the two algorithms are present. Finally, a comparison of graphical, numerical, and proposed software is illustrated for the said maps. The newly suggested MS-Dos software allows mathematicians to determine the above behavior of various one-dimensional real maps except for any complicated code.

Developing Dynamical Programming Software (DPS.exe)

The first requirement is to introduce the works, for example, a flowchart to classify functions’ essence to explore the dynamical simulation framework. The diagram (Figure 1) depicts the developing technique and application of the process of the newly proposed DPS.exe.

Case-I: One-dimensional first-degree map

The general form of the one-dimensional first-degree equation is $y=f\left(x\right)=$ $ax+b$, where a and b are the real constants, and x is the variable.

Fixed point analysis

A specific value of $a,b$ the FORTRAN command [44] gives the output Figure 2(i). But in this case, if b = 0 and x = 1, then $f\left(x\right)=x$ and thus all the

Figure 1. Working procedure of developing DPS.exe.

Figure 2. Dynamical behavior of first-degree map function type (ii) value and nature of the fixed point (iii) orbit diagram (iv) fate of the orbit (v) process of the new interface.

points of $f\left(x\right)$ will be the fixed points, which is $x=\frac{b}{1-a}\left(a\ne 1\right)$ . This condition will be $\left(a\ne 1\right)$ overcome using the IF statement in the first line in FORTRAN command [44] . Again, when $f\left(x\right)=y=x$ there is no fixed point, i.e., the parallel lines meet at infinity. If $\frac{b}{1-a}>10000000$ or $\frac{b}{1-a}<-10000000$ (this range can be changed for more reliable calculation), the line is eventually parallel, so no fixed point exists. Nevertheless, if $-1000000<\frac{b}{1-a}<1000000$ (under consideration), then there must be a fixed point presented by $sct=\frac{b}{1-a}$ . The nature of the fixed point [16] (attracting, repelling, or neutral) will be determined using the following conditions:

$x_0=sct=\frac{b}{1-a}$ is $\{\begin{array}{l}\text{attracting}\text{;}\text{if}\left|f^{\prime }\left(x_0\right)\right|<1\\ \text{repelling}\text{;}\text{if}\left|f^{\prime }\left(x_0\right)\right|>\text{1}\\ \text{neutral}\text{;}\text{if}\left|f^{\prime }\left(x_0\right)\right|=1\end{array}$ .

The nature of the fixed point entirely depends on the value of a as $f^{\prime }\left(x\right)=a$ , $f^{\prime }\left(x_0\right)=a$ . Therefore, the output of the FORTRAN executable (DPS.exe) interface Figure 2(ii).

Orbit analysis

Let $x=x_0$ be the initial seed. Then the orbit analysis of $f\left(x\right)=ax+b$ is $x_0\to f\left(x_0\right)=a{x}_0+b$ , $f^2\left(x_0\right)=f\left(f\left(x_0\right)\right)=a\left(a{x}_0+b\right)+b=a^2{x}_0+ab+b$ and so on.

The output of this segment for $f\left(x\right)=2x-3$ with the initial seed $x_0=3.5$ is presented in Figure 2(iii).

Fate of orbit

After continuing the iteration process sufficiently many more times, finally, the fate of the orbit is presented in Figure 2(iv). Now, users may need to repeat the process for any new function. This programming procedure automatically returns to the initial stage Figure 2(v). This section’s output proceeds the mathematician to the end of a program or the program’s initial phase.

Case-II: One-dimensional second-degree map

The general form of the one-dimensional second-degree equation is $y=f\left(x\right)=a{x}^2+bx+c$ , where a, b and c are the real constants and x is the variable.

Fixed point analysis

The specific values of $a,b,c$ FORTRAN [46] give the output Figure 3(i).

Fixed point of $f\left(x\right)=a{x}^2+bx+c$ is $x=\frac{-\left(b-1\right)\pm \sqrt{{\left(b-1\right)}^2-4ac}}{2a}$ .

When ${\left(b-1\right)}^2-4ac>0$ then two fixed points exist, and those two fixed points are

$\frac{-\left(b-1\right)+\sqrt{{\left(b-1\right)}^2-4ac}}{2a}$ and $\frac{-\left(b-1\right)-\sqrt{{\left(b-1\right)}^2-4ac}}{2a}$

Figure 3. Dynamical behavior of second-degree map (i) function type (ii) value and nature of the fixed point (iii) orbit diagram.

Therefore, the output of the FORTRAN executable (DPS.exe) file is pictured in Figure 3(ii).

Orbit analysis

The output of this segment $f\left(x\right)$ with the initial seed $x_0=1.7$ demonstrated in Figure 3(iii).

Case-III: One-dimensional third-degree maps

The general form of the one-dimensional third-degree equation is $y=f\left(x\right)=$ $a{x}^3+b{x}^2+cx+d$ where a, b, c, and d are the real constants and x is the variable.

Fixed point analysis

For a specific value of $a,b,c,d$ FORTRAN command [46] generates the Figure 4(i).

Root process for finding fixed points

The fixed point is the point of intersection of $y=f\left(x\right)$ and $y=x$ . Using Mathematica or any other programming command [41] - [45] , one can find its fixed points. As it is complicated and lengthy, the numerical procedure may help to obtain the solution.

Numerical process for finding fixed points

The solution of finding the given equation’s solution is to set an initial value of x. This value maybe −10,000 or less. Now choose $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ , $g\left(x\right)=x$ .

If $f\left(x\right)=g\left(x\right)$ , then x is a fixed point, start checking with −10,000. If both $f\left(x\right)$ and $g\left(x\right)$ are not equal, then do the process for $x=-10000+0.00001$

Similarly, if it is not equal yet, then do it again for $x=-10000.00001+0.00001$

All these procedures can be quickly done using the FORTRAN command [46] . If any fixed point can be found, then the nature of the fixed point can be determined by the logic of $\left|f^{\prime }\left(x_0\right)\right|$ .

For the specific function $f\left(x\right)=x^3+2x^2-3x+4$ , the output is revealed in

Figure 4. Dynamical behavior of cubic map (i) function type (ii) value and nature of the fixed point (iii) orbit diagram.

Figure 4(ii).

Orbit analysis and the fate of the orbit

If the iterative value for any specific function goes to <−10¹² or goes to >10¹² after some iterations, then the fate of the orbit goes to negative infinity or positive infinity, respectively. For the specific function $f\left(x\right)$ with the initial seed $x_0=-3.3$ , the output is visualized in Figure 4(iii).

Case-IV: One-dimensional higher degree maps

One-dimensional higher degree equation can be expressed in the following form:

$f\left(x\right)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_n{x}^n$

where $a_0,a_1,a_2,\cdots ,a_n$ are n numbers of coefficients and x is the variable. Developing DPS.exe for the one-dimensional higher degree function is more complicated, as described in the later section.

Equation generating technique

A glance at the development of DPS.exe for one-dimensional higher-degree maps has been described in Figure 5.

Figure 5. Generating process of one-dimensional n^th-degree function.

Suppose anyone is interested to know the dynamical behavior of a function of the 5^th degree. Then executing this part of programming displays Figure 6(i) and needs to input the value of n as 5. Here the number of the variable associated with each term depends on the desire of any individual. So there needs to build an array of variables A Figure 6(ii) such as A (10,000). Now the focus is on the value of those variables. Anyone needs to input the values of variables for any specific degree function. Inputting the values of the associated variables, visualize the complete process and associated programming code stored in DPS.exe engine code. Then for particular values $n=5$ and coefficients, $A_0=2$ , $A_1=-1$ , $A_2=3$ , $A_3=-4$ , $A_4=1$ , $A_5=7$ , the entire function is demonstrated in Figure 6(iii).

Numerical procedure obtaining fixed point

This procedure is identical to the numerical process of the third-degree equation, the generalized form of $f^{\prime }\left(x\right)$ is $f^{\prime }\left(x\right)={\displaystyle \underset{n=0}{\overset{n-1}{\sum }}n{a}_n{x}^{n-1}}$ . For the specific 5^th-degree function, the output of the following programming segment is portrayed in Figure 6(iv).

Orbit analysis and the fate of the orbit

This procedure is equivalent to the third-degree equation, and the output of this segment of engine code unfolds in Figure 6(v).

Case-V: Experiment on Higher degree function

Here, the 6^th-degree equation $f\left(x\right)=3+2x-7x^2-1.5x^3+2.2x^4-3.7x^5+1.1x^6$ has been considered. Then DPS.exe exhibits the dynamical info in Figure 7(i) an Figure 7(ii). To see the orbit for any specific initial seed, press 5, input the number of iterations (12, but it depends on the user’s desire), and the initial seed’s value ( $x_0=3.1$ ). The desired interface is in Figure 7(iii).

Case-VI: Exponential maps

The generalized form of an exponential map is $f\left(x\right)=a{\text{e}}^{bx}+c$ , where $a,b,c$ are the arbitrary constants. The following source code asks the user of DPS.exe for specific values of $a,b,c$ and finally expresses the function. The process of

Figure 6. Dynamical behavior of n^th degree (n = 5) map (i) choosing the degree of function (ii) coefficient value inputting (iii) final equation (iv) value and nature of the fixed point (v) orbit diagram and fate.

Figure 7. Dynamical behavior of n^th degree (n = 6) map (i) choosing the degree of function (ii) coefficient value inputting, final equation, value, and nature of the fixed point (iii) orbit diagram and fate.

finding the fixed point, nature of the fixed point, orbits, and fate of the orbit of $f\left(x\right)$ under a specific initial seed is the same as mentioned previously. The dynamical behavior of $f\left(x\right)=3{\text{e}}^{1.1x}-2$ appears in Figure 8(i). After pressing 5, the system will represent the orbit analysis for any particular seed $x_0=-0.9$ in Figure 8(ii). Analogously, anyone can determine any exponential functions dynamical behavior by changing the value of coefficients.

Case-VII: Logarithmic maps

The generalized form of the exponential map is $f\left(x\right)=a\log \left(bx\right)+c$ , where $a,b,c$ are arbitrary constants. The dynamical behavior of $f\left(x\right)=2\log \left(3x\right)+4$ appeared in Figure 9(i) an Figure 9(ii). Here, the number of iteration is 15, and the initial seed is 5.

Case-VIII: Absolute value maps

The generalized form of the absolute value map is $f\left(x\right)=\left|a{x}^b\right|+c$ , where $a,b,c$ are arbitrary constants. The dynamical behavior of $f\left(x\right)=\left|-2x^3\right|-4$ demonstrated in Figure 10.

3. Result and Discussions

Exploring the exactness of the obtained result using DPS.exe has to compare it numerically and graphically. In numerical cases, the initial seed’s specific value

Figure 8. Dynamical behavior of exponential map (i) coefficient value inputting, final equation, value, and nature of the fixed points (ii) orbit diagram of different iterations.

Figure 9. Dynamical behavior of logarithmic map (i) coefficient value inputting, final equation, value, and nature of the fixed points (ii) orbit diagram of different iterations.

Figure 10. Dynamical behavior of absolute value map.

gives the following values $f\left(x\right)$ , $f^2\left(x\right)=f\left(f\left(x\right)\right)$ , $f^3\left(x\right)=f\left(f\left(f\left(x\right)\right)\right)$ , $\cdots$ etc. On the other hand, graphical analysis shows the graph of a function, fixed point, and orbit of the fixed point under a specific initial seed. It is more apparent to determine the dynamical behavior from its graphical analysis. However, DPS.exe has been more straightforward for a mathematician to gather all the information about dynamical behavior without programming knowledge.

3.1. One-Dimensional First-Degree Map

Suppose the one-dimensional first-degree equation is $f\left(x\right)=2x+1$ and the initial seed $x_0$ .

Numerical analysis

Therefore, $f\left(x\right)=2x+1$ , $f^2\left(x\right)=4x+3$ , $f^3\left(x\right)=8x+7$ , $f^4\left(x\right)=16x+5$ , $\cdots$ , and so on. $f\left(x\right)=x$ gives the desired fixed point, and the fixed point is $x=-1$ . This fixed point is repelling because if $x_0=-1.1$ that is a nearby point of the initial seed, then the orbit of the function appears as follows:

$-1.1\to -1.2\to -1.4\to -1.8\to \cdots$ and so on.

Thus, the orbit of the function under the considered initial seed is −∞.

Graphical Analysis

The graphical representation of $f\left(x\right)=2x+1$ and its dynamical behavior ensues in Figure 11.

Here, the orbit of the point for the given function is repelling, represented by the blue staircase.

DPS.exe analysis

In this process, mathematicians need not apply any programming command, just run DPS.exe and insert the coefficients. As $f\left(x\right)$ is a linear function, after clicking DPS.exe, press 1 for the linear function section, which is exhibited in the appendix (A-I Figure A1.1).

Now for the function $f\left(x\right)$ insert “2” as the value of “a”, and “1” as “b”. The computer will then do the rest of the job to determine all dynamical behavior, displayed in the appendix (A-I Figure A1.2).

DPS.exe also offers to see the orbit of the function for any desired initial seed. For this, the user has to press “5” and enter. Then, insert the initial seed and the number of iterations. Finally, the appendix demonstrates the interface (A-I Figure A1.3).

Finally, all comparisons of one-dimensional first-degree maps, namely, numerical, graphical, and DPS.exe are presented in Table 1.

3.2. One-Dimensional Second-Degree Map

Suppose the one-dimensional second-degree equation is $f\left(x\right)=\frac{1}{2}{x}^2-1$ . Then,

Figure 11. Graphical representation of one-dimensional first-degree map.

as previously, the DPS.exe interface is displayed in the appendix (A-II Figures A2.1-A2.3), and all comparisons of one-dimensional second-degree maps, namely, numerical, graphical, and DPS.exe are presented in Table 2.

3.3. Higher Degree Maps

Suppose the one-dimensional higher (4^th) degree map is $f\left(x\right)=2x^4-3x^3-4x^2-5x-7$ . The DPS.exe interface is manifested in the appendix (A-III Figures A3.1-A3.4), and all comparisons of one-dimensional higher-degree maps are presented in Table 3.

Table 1. Comparison between numerical, graphical, and DPS.exe analysis.

Table 2. Comparison between numerical, graphical, and DPS.exe analysis.

Table 3. Comparison between numerical, graphical, and DPS.exe analysis.

Table 4. Comparison between numerical, graphical, and DPS.exe analysis.

3.4. Exponential Maps

Suppose the exponential function is $f\left(x\right)=\frac{1}{2}{\text{e}}^{2x}-2$ . Therefore, the DPS.exe interface is demonstrated in the appendix (A-IV Figures A4.1-A4.3), and all comparisons of the exponential map are presented in Table 4.

4. Conclusions

The one-dimensional real map is perceived as difference equations, iterated maps, or recursion relations in mathematical systems that model a single variable due to evolving over discrete steps. It has a remarkable significance in modeling natural phenomena, for example, population dynamics, electronics, and economics. However, this study has profoundly elaborated a possible one-dimensional real maps coding system to know the dynamical behavior and proposed a new technique, executable dynamical programming software in short DPS.exe. The appropriateness of the proposed DPS.exe is then systematically investigated graphically and numerically.

The present work conducted a theoretical, graphical, and extensive numerical analysis to comprehensively explore one-dimensional real maps of dynamical behavior: first-degree, second-degree, third-degree, nth-degree, exponential, logarithmic, and absolute. The main focus is on one-dimensional real maps to demonstrate dynamic behavior in the system. A sensible relationship between the graphical, numerical, and DPS.exe has drowned. Furthermore, DPS.exe is an effective software for determining the dynamical behavior of one-dimensional real maps rather than general calculating, Mathematica, or other programming languages. This analytical research suggests that the newly proposed MS-Dos software allows mathematicians and physicists to determine various one-dimensional real maps’ dynamical behavior without complicating programming code. We plan to analyze the chaotic maps using the current mechanism.

Acknowledgements

The authors thank the anonymous reviewers for their suggestions and invaluable comments.

Author Contributions

Each author equally contributed to this paper and read and approved the final manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

Appendix A

A-I: One-dimensional first-degree map

Figure A1.1. Identifying the desired function.

Figure A1.2. Identifying desired function’s fixed point.

Figure A1.3. Identifying desired function’s nature of the fixed point.

A-II: One-dimensional second-degree map

Figure A2.1. Identifying the desired function.

Figure A2.2. Identifying desired function’s fixed points.

Figure A2.3. Identifying desired function’s nature of the fixed points.

A-III: Higher degree maps

Figure A3.1. Identifying the desired function.

Figure A3.2. Inputting desired function’s associated co-efficient.

Figure A3.3. Identifying desired function’s fixed points and nature.

Figure A3.4. Identifying desired function’s fate of the orbit.

A-IV: Exponential maps

Figure A4.1. Identifying the desired function.

Figure A4.2. Inputting desired function’s associated co-efficient and identifying fixed points & nature.

Figure A4.3. Identifying desired function’s fate of the orbit.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Newton, I. (1729) The Mathematical Principles of Natural Philosophy (Vol. 2). 3rd Edition, William Dawson and Sons, London. https://www.google.com.hk/books/edition/The_Mathematical_Principles_of_Natural_P/Tm0FAAAAQAAJ?hl=en&gbpv=0
[2]	Poincaré, H. (1967) New Methods of Celestial Mechanics. American Institute of Physics, Melville.
[3]	Poincaré, H. (1890) Chapitre I. Propriétésgénérales des équationsdifférentielles. Acta Mathematica, 13, 8-45. https://doi.org/10.1007/bf02392507
[4]	Julia, G. (1918) Mémoiresurl’iteration des fonctionsrationnelles. Journal de Mathématiques Pureset Appliquées, 8, 47-245.
[5]	Fatou, P. (1917) Sur les substitutions rationnelles. Comptes Rendusdel Académie des Sciences de Paris, 164, 806-808.
[6]	Birkhoff, G.D. (1920) Recent Advances in Dynamics. Science, 51, 51-55. https://doi.org/10.1126/science.51.1307.51
[7]	Smale, S. (1999) Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations. Journal of Nonlinear Science, 9, 363-415. https://doi.org/10.1007/s003329900074
[8]	Lorenz, E.N. (1972) Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas. American Association for the Advancement of Science, Washington DC.
[9]	Robert, M.M. (1976) Simple Mathematical Models with Very Complicated Dynamics. Nature, 261, 459-467. https://doi.org/10.1038/261459a0
[10]	Mandelbrot, B.B. (1983) The Fractal Geometry of Nature. American Journal of Physics, 51, 286-287. https://doi.org/10.1119/1.13295
[11]	Douady, A. and Hubbard, J.H. (1984) Étude dynamique des polynômes complexes. Université de Paris-Sud, Paris. https://pi.math.cornell.edu/~hubbard/OrsayFrench.pdf
[12]	Sullivan, J.M. (2012) Mathematical Pictures: Visualization, Art and Outreach. In: Behrends, E., Crato, N. and Rodrigues, J., Eds., Raising Public Awareness of Mathematics, Springer, Berlin, 279-293. https://doi.org/10.1007/978-3-642-25710-0_21
[13]	de Melo, W. and van Strien, S. (1992) One-Dimensional Dynamics. Springer eBooks. https://doi.org/10.1007/978-3-642-78043-1
[14]	Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G. and Fedorenko, V.V. (1997) Dynamics of One-Dimensional Maps. Springer, New York. https://doi.org/10.1007/978-94-015-8897-3
[15]	Rodrigues, A. (2021) One-Dimensional Dynamical Systems: An Example-Led Approach. Chapman and Hall/CRC Press, New York. https://doi.org/10.1201/9781003144618
[16]	Grandmont, J.M. (2021) Cycles and Chaos in Economic Equilibrium: Periodic and Aperiodic Behaviour in Discrete One-Dimensional Dynamical Systems. Princeton University Press eBooks, 44-63. https://doi.org/10.2307/j.ctv19fvxt1.5
[17]	Sharkovsky, A.N. (2014) On the History of One-Dimensional Dynamics. ESAIM: Proceedings and Surveys, 46, 83-85. https://doi.org/10.1051/proc/201446007
[18]	Elaydi, S. (2019) Global Dynamics of Discrete Dynamical Systems and Difference Equations. In: Elaydi, S., Pötzsche, C. and Sasu, A., Eds., ICDEA 2017: Difference Equations, Discrete Dynamical Systems and Applications, Springer, Cham, 51-81. https://doi.org/10.1007/978-3-030-20016-9_3
[19]	Tanaka, S. (2019) Brain as a Dynamical System. Journal of Brain and Nerves, 71, 657-664.
[20]	Mehran, N.Z., Panahi, S., Hosseini, Z., Golpayegani, S.M.R.H. and Jafari, S. (2020) One Dimensional Map-Based Neuron Model: A Phase Space Interpretation. Chaos, Solitons & Fractals, 132, Article ID: 109558. https://doi.org/10.1016/j.chaos.2019.109558
[21]	Clark, T., Strien, S.V. and Trejo, S. (2013) Complex Box Bounds for Real Maps. Communications in Mathematical Physics, 355, 1001-1119. https://doi.org/10.1007/s00220-017-2958-y
[22]	Iwanaga, S. and Namatame, A. (2016) Contagion of Evacuation Decision Making on Real Map. Mobile Networks and Applications, 21, 206-214. https://doi.org/10.1007/s11036-016-0704-x
[23]	Jia, X., Yujun, K., Enzhan, Z. and Weili, J. (2011) Research on Mobility Model for VANETs Based on Real Map. International Conference on Computational Problem-Solving (ICCP), Chengdu, 21-23 October 2011, 1-6. https://doi.org/10.1109/ICCPS.2011.6089939
[24]	Joshi, Y. and Blackmore, D. (2012) Exponentially Decaying Discrete Dynamical Systems. Recent Patents on Space Technology, 2, 37-48. https://doi.org/10.2174/1877611611202010037
[25]	Kozlovski, O. (2013) Periodic Attractors of Perturbed One-Dimensional Maps. Ergodic Theory and Dynamical Systems, 33, 1519-1541. https://doi.org/10.1017/etds.2013.28
[26]	Rivera-Letelier, J.R. and Shen, W. (2014) Statistical Properties of One-Dimensional Maps under Weak Hyperbolicity Assumptions. arXiv: 1004.0230.
[27]	Bruin, H. and Vejnar, B. (2020) Classification of One Dimensional Dynamical Systems by Countable Structures. arXiv: 2006.14926.
[28]	Ohmori, S. and Yamazaki, Y. (2020) Ultradiscrete Bifurcations for One Dimensional Dynamical Systems. Journal of Mathematical Physics, 61, Article ID: 122702. https://doi.org/10.1063/5.0012772
[29]	Sushko, I., Gardini, L. and Avrutin, V. (2016) Non-Smooth One-Dimensional Maps: Some Basic Concepts and Definitions. Journal of Difference Equations and Applications, 22, 1816-1870. https://doi.org/10.1080/10236198.2016.1248426
[30]	Li, Q., Tang, S. and Feng, X. (2012) Computing 1D Discontinuous Boundaries of Dynamical Systems. 2012 24th Chinese Control and Decision Conference (CCDC), Taiyuan, 23-25 May 2012, 1427-1430. https://doi.org/10.1109/CCDC.2012.6244229
[31]	Ufuktepe, Ü. (2014) Applications of Discrete Dynamical Systems with Mathematicä. Conference: RIMSAt, Kyoto, February 2014.
[32]	Jorba, À., Rabassa, P. and Tatjer, J.C. (2016) Local Study of a Renormalization Operator for 1D Maps under Quasiperiodic Forcing. Discrete & Continuous Dynamical Systems, 9, 1171-1188. https://doi.org/10.3934/dcdss.2016047
[33]	Khmou, Y., Said, S. and Frikel, M. (2018) A Comparison of Entropy Metrics in 1D Discrete Dynamical System. 1st International Conference on Signals, Automation and Telecommunications, Beni Mellal, 2-4 May 2018.
[34]	Bashkirtseva, I. and Tsvetkov, I. (2018) Impact of the Parametric Noise on Map-Based Dynamical Systems. AIP Conference Proceedings, 2025, Article ID: 040004. https://doi.org/10.1063/1.5064888
[35]	Ballard, T., Palada, H., Griffin, M. and Neal, A. (2019) An Integrated Approach to Testing Dynamic, Multilevel Theory: Using Computational Models to Connect Theory, Model, and Data. Organizational Research Methods, 24, 251-284. https://doi.org/10.1177/1094428119881209
[36]	Medrano, F.F. and Solis, F.J. (2015) Stability of Real Parametric Polynomial Discrete Dynamical Systems. Discrete Dynamics in Nature and Society, 2015, Article ID: 680970. https://doi.org/10.1155/2015/680970
[37]	Bai, Y., Bilige, S. and Chaolu, T. (2018) Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations. Journal of Applied Mathematics and Physics, 6, 1825-1839. https://doi.org/10.4236/jamp.2018.69156
[38]	Xiang, C. and Wang, H. (2019) Ground-State Energy and Entropy for One-Dimensional Heisenberg Chain with Alternating D-Term. Journal of Applied Mathematics and Physics, 7, 1220-1225. https://doi.org/10.4236/jamp.2019.75082
[39]	Devaney, R.L. and Choate, J. (2000) Chaos: A Tool Kit of Dynamics Activities. Key Curriculum Press, Emeryville.
[40]	Layek, G.C. (2015) An Introduction to Dynamical Systems and Chaos. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2556-0
[41]	Lynch, S. (2007) Dynamical Systems with Applications Using Mathematica. Birkhauser, New York.
[42]	Kulenovic, M.R.S. and Merino, O. (2002) Discrete Dynamical Systems and Difference Equations with Mathematica. CRC Press, New York.
[43]	Lynch, S. (2004) Dynamical Systems with Applications Using MATLAB. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8156-2
[44]	Lynch, S. (2010) Dynamical Systems with Applications Using Maple. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-4605-9
[45]	Nicola, B., Luigi, P. and Antonio, R. (2000) Mechanics and Dynamical Systems with Mathematica. Springer, New York.
[46]	Lipschutz, S. and Poe, A. (1982) Programming with FORTRAN. McGraw-Hill, Singapore.