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.
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.
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.