Research Article
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Dynamical Interpretation of Logistic Map using Euler’s Numerical Algorithm

Year 2022, Volume 4, Issue 3, 128 - 134, 30.11.2022
https://doi.org/10.51537/chaos.1164683

Abstract

In the last two decades, the dynamics of difference and differential equations have found a celebrated place in science and engineering such as weather forecasting, secure communication, transportation problems, biology, the population of species, etc. In this article, we deal with the dynamical behavior of the logistic map using Euler’s numerical algorithm. The dynamical properties of Euler’s type logistic system are derived analytically as well as experimentally. In the analytical section, the dynamical properties such as fixed point, period-doubling, and irregularity are examined followed by s few theorems. Further, in the experimental section, the dynamical properties of Euler’s type logistic system are studied using period-doubling bifurcation plots. Because the dynamics of the Euler’s map depend on the Euler’s control parameter h, therefore, three major cases are discussed for all the dynamical properties for h = 0.1, 0.4, and 0.7. The result shows that as the value of parameter h decreases from 1 to 0 the growth rate parameter r increases rapidly. Therefore, the improved chaotic regime in bifurcation plots may improve the chaos based applications in science and engineering such as secure communication.

References

  • Alligood, K. T., Sauer, T. D. and Yorke, J. A., 1996 Chaos : An Introduction to Dynamical Systems, Springer Verlag, New York Inc.
  • Ashish, Cao, J. and Chugh, R., 2018 Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model, Nonlinear Dynamics 94: 959-975.
  • Ashish and Cao, J., 2019a A novel fixed point feedback approach studying the dynamcial behaviour of standard logistic map, International Journal of Bifurcation and Chaos 29: 1950010-16, 16 pages.
  • Ashish, Cao, J., Alsaadi, F. and Malik, A. K., 2021a Discrete Superior Hyperbolicity in Chaotic Maps, Chaos Theory and Applications 03: 34-42
  • Ashish, Cao, J. and Chugh, R., 2021b Discrete chaotification in modulated logistic system, International Journal of Bifurcation and Chaos 31: 2150065, 14 Pages.
  • Ashish, Cao, J. and Chugh, R., 2019b Controlling chaos using superior feedback technique with applications in discrete traffic models, International Journal of Fuzzy System 21: 1467-1479.
  • Ashish, Cao, J. and Alsaadi, F., 2021c Chaotic evolution of difference equations in Mann orbit, J. Appl. Anal. Comput. 11: 1-20.
  • Ausloos, M. and Dirickx M., 2006 The Logistic Map and the Route to Chaos : from the Beginnings to Modern Applications, Springer Verlag, New York Inc.
  • Chugh, R., Rani, M. and Ashish, 2012 Logistic map in Noor orbit, Chaos and Complexity Letter 6, 167-175.
  • Devaney, R. L., 1948 An Introduction to Chaotic Dynamical Systems, 2nd Edition (Addison-Wesley).
  • Feigenbaum, M. J., 1978 Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19: 25-52.
  • He, J. H., Jiao, M. L., Gepreel, K. A. and Khan, Y., 2023 Homotopy perturbation method for strongly nonlinear oscillators, Mathematics and Computers in Simulation 204: 243-258.
  • Holmgren, R. A., 1994 A First Course in Discrete Dynamical Systems, Springer Verlag, New York Inc.
  • Khamosh, Kumar, V. and Ashish, 2020 A novel feedback control system to study the stability in stationary states, Journal of Mathematical and Computational Science 10: 2094-2109.
  • Kumar, M. and Rani, M., 2005 An experiment with summability methods in the dynamics of logistic model, Indian J Math 47: 77-89.
  • Kumar, V., Khamosh and Ashish, 2020 An empirical approach to study the stability og generalized logistic map in superior orbit, Advances In Mathematics: Scientific Journal 10: 2094-2109.
  • Lorenz, E. N., 1963 Deterministic nonperiodic flows, J. Atoms. Sci. 20: 130-141.
  • Mann,W. R., 1953 Mean value methods in iteration, Proceedings of American Mathematical Society 04: 506-510.
  • May, R., 1976 Simple mathematical models with very complicated dynamics, Nature 261, 459-475.
  • Molina, C., Sampson N., Fitzgerald W. J. and Niranjan M., 1996 Geometrical techniques for finding the embedding dimension of time series, Proc. IEEE Signal Processing SocietyWorkshop, 161-169.
  • Parasad, B. and Katiyar, K., 2014 A stability analysis of logistic model, Int. J. Non-Lin Sci. 17: 71-79.
  • Poincare H., 1899 Les Methods Nouvells de la Mecanique Leleste, Gauthier Villars, Paris.
  • Rani, M. and Kumar, V., 2005 A new experimental approach to study the stability of logistic maps, J. Indian Acad. Math. 27: 143-156.
  • Rani, M. and Agarwal, R., 2009 A new experiment approach to study the stability of logistic map, Chaos Solitions Fractals 04: 2062-2066.
  • Radwan, A. G., 2013 On Some Generalized Discrete Logistic Map, J. Adv. Research 04: 163-171.
  • Renu, Ashish, and Chugh, R., 2022 On the dynamics of a discrete difference map in mann orbit, Computational and Applied Mathematics 41: 1-19.
  • Robinson, C., 1995 Dynamical Systems: Stabilily, Symbolic Dynamics, and Chaos, CRC Press.
  • Singh, N. and Sinha, A., 2010 Chaos based secure communication system using logistic map, Optics and Lasers in Engineering 48: 398-404.
  • Song, N. and Meng, J., 1996 Research on Logistic mapping and Synchronization, Proc. IEEE Intell. Contr. Autom. 01: 987-991.

Year 2022, Volume 4, Issue 3, 128 - 134, 30.11.2022
https://doi.org/10.51537/chaos.1164683

Abstract

References

  • Alligood, K. T., Sauer, T. D. and Yorke, J. A., 1996 Chaos : An Introduction to Dynamical Systems, Springer Verlag, New York Inc.
  • Ashish, Cao, J. and Chugh, R., 2018 Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model, Nonlinear Dynamics 94: 959-975.
  • Ashish and Cao, J., 2019a A novel fixed point feedback approach studying the dynamcial behaviour of standard logistic map, International Journal of Bifurcation and Chaos 29: 1950010-16, 16 pages.
  • Ashish, Cao, J., Alsaadi, F. and Malik, A. K., 2021a Discrete Superior Hyperbolicity in Chaotic Maps, Chaos Theory and Applications 03: 34-42
  • Ashish, Cao, J. and Chugh, R., 2021b Discrete chaotification in modulated logistic system, International Journal of Bifurcation and Chaos 31: 2150065, 14 Pages.
  • Ashish, Cao, J. and Chugh, R., 2019b Controlling chaos using superior feedback technique with applications in discrete traffic models, International Journal of Fuzzy System 21: 1467-1479.
  • Ashish, Cao, J. and Alsaadi, F., 2021c Chaotic evolution of difference equations in Mann orbit, J. Appl. Anal. Comput. 11: 1-20.
  • Ausloos, M. and Dirickx M., 2006 The Logistic Map and the Route to Chaos : from the Beginnings to Modern Applications, Springer Verlag, New York Inc.
  • Chugh, R., Rani, M. and Ashish, 2012 Logistic map in Noor orbit, Chaos and Complexity Letter 6, 167-175.
  • Devaney, R. L., 1948 An Introduction to Chaotic Dynamical Systems, 2nd Edition (Addison-Wesley).
  • Feigenbaum, M. J., 1978 Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19: 25-52.
  • He, J. H., Jiao, M. L., Gepreel, K. A. and Khan, Y., 2023 Homotopy perturbation method for strongly nonlinear oscillators, Mathematics and Computers in Simulation 204: 243-258.
  • Holmgren, R. A., 1994 A First Course in Discrete Dynamical Systems, Springer Verlag, New York Inc.
  • Khamosh, Kumar, V. and Ashish, 2020 A novel feedback control system to study the stability in stationary states, Journal of Mathematical and Computational Science 10: 2094-2109.
  • Kumar, M. and Rani, M., 2005 An experiment with summability methods in the dynamics of logistic model, Indian J Math 47: 77-89.
  • Kumar, V., Khamosh and Ashish, 2020 An empirical approach to study the stability og generalized logistic map in superior orbit, Advances In Mathematics: Scientific Journal 10: 2094-2109.
  • Lorenz, E. N., 1963 Deterministic nonperiodic flows, J. Atoms. Sci. 20: 130-141.
  • Mann,W. R., 1953 Mean value methods in iteration, Proceedings of American Mathematical Society 04: 506-510.
  • May, R., 1976 Simple mathematical models with very complicated dynamics, Nature 261, 459-475.
  • Molina, C., Sampson N., Fitzgerald W. J. and Niranjan M., 1996 Geometrical techniques for finding the embedding dimension of time series, Proc. IEEE Signal Processing SocietyWorkshop, 161-169.
  • Parasad, B. and Katiyar, K., 2014 A stability analysis of logistic model, Int. J. Non-Lin Sci. 17: 71-79.
  • Poincare H., 1899 Les Methods Nouvells de la Mecanique Leleste, Gauthier Villars, Paris.
  • Rani, M. and Kumar, V., 2005 A new experimental approach to study the stability of logistic maps, J. Indian Acad. Math. 27: 143-156.
  • Rani, M. and Agarwal, R., 2009 A new experiment approach to study the stability of logistic map, Chaos Solitions Fractals 04: 2062-2066.
  • Radwan, A. G., 2013 On Some Generalized Discrete Logistic Map, J. Adv. Research 04: 163-171.
  • Renu, Ashish, and Chugh, R., 2022 On the dynamics of a discrete difference map in mann orbit, Computational and Applied Mathematics 41: 1-19.
  • Robinson, C., 1995 Dynamical Systems: Stabilily, Symbolic Dynamics, and Chaos, CRC Press.
  • Singh, N. and Sinha, A., 2010 Chaos based secure communication system using logistic map, Optics and Lasers in Engineering 48: 398-404.
  • Song, N. and Meng, J., 1996 Research on Logistic mapping and Synchronization, Proc. IEEE Intell. Contr. Autom. 01: 987-991.

Details

Primary Language English
Subjects Mathematics, Interdisciplinary Applications
Journal Section Research Articles
Authors

Sanjeev . This is me
Suraj Degree College
0000-0002-2263-3525
India


Anjali . This is me
National Institute of Technology Mizoram
0000-0002-9296-9542
India


Ashish ASHİSH> (Primary Author)
Government College Satnali, Mahendergarh, INDIA
0000-0001-9598-3393
India


A. K. MALİK This is me
UP Rajarshi Tandon Open University, Prayagraj
0000-0002-1520-0115
India

Publication Date November 30, 2022
Published in Issue Year 2022, Volume 4, Issue 3

Cite

APA ., S. , ., A. , Ashish, A. & Malik, A. K. (2022). Dynamical Interpretation of Logistic Map using Euler’s Numerical Algorithm . Chaos Theory and Applications , 4 (3) , 128-134 . DOI: 10.51537/chaos.1164683

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830