[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamics of $q$-Deformed Logistic Map via Superior Approach

Journal of Applied Nonlinear Dynamics 12(2) (2023) 285--296 | DOI:10.5890/JAND.2023.06.007

Renu$^{1}$, Ashish$^{2}$, Renu Chugh$^{1}$

$^{1}$ Department of Mathematics, Maharshi Dayanand University, Rohtak-124001, Haryana, India

$^{2}$ Department of Mathematics, Government College Satnali, Mahendergarh-123024, Haryana, India

Download Full Text PDF

 

Abstract

Inspired by various studies on $q$-deformations in physical systems and their wider applications in different fields, we study the $q$-deformed logistic map via a superior fixed-point feedback approach. Different dynamical characteristics such as fixed-points, time-series evolution, period-doubling bifurcation, and Lyapunov exponent of the system are analyzed using a superior approach. The results are carried out analytically as well as experimentally. Under the proposed approach, due to the presence of an additional control parameter $\alpha$, the system exhibits superior dynamical characteristics such as better stability range, suitable lower maximum Lyapunov exponent value, and an improved sensitivity as compared to the traditional approach.

Acknowledgments

This work is supported by the University Grants Commission of India under Grant No. (F.No. $16$-$6$(DEC. $2017$)/$2018$(NET/CSIR) UGC Ref. No.: 1049/(CSIR-UGC NET DEC. 2017)).

References

  1. [1] Chaichian, C. and Demichev, A. (1996), Introduction to Quantum groups, World Scientific: Singapore.
  2. [2] Banerjee, S. and Parthasarathy, R. (2011), A $q$-deformed logistic map and its implications, Journal of Physics A: Mathematical and Theoretical, 44(04), 04510.
  3. [3] Devaney, R.L. (1992), A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley: Boston.
  4. [4] Devaney, R.L. (1948), An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley: Boston.
  5. [5] Holmgren, R.A. (1994), A First Course in Discrete Dynamical Systems, Springer: New York.
  6. [6] Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996), Chaos: An Introduction to Dynamical Systems, Springer: New York.
  7. [7] Ausloos, M. and Dirickx, M. (2006), The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications, Springer: New York.
  8. [8] Strogatz, S. H. (1994), Nonlinear Dynamics and Chaos, Persus Books Publishing: L.L.C., New York.
  9. [9] Robinson, C. (1995), Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press: Boca Raton.
  10. [10] Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamics and Chaos, Springer: New York.
  11. [11] Elagdi, S.N. (1999), Chaos: An Introduction to Difference Equations, Springer: New York.
  12. [12] Elhadj, Z. and Sprott, J.C. (2008), The effect of modulating a parameter in the logistic map, Chaos, 18(2), 1–7.
  13. [13] Martelli, M. (1999), Chaos: An Introduction to Discrete Dynamical Systems and Chaos, Wiley-Interscience Publication: New York Inc..
  14. [14] Diamond, P. (1976), Chaotic behaviour of systems of difference equations, International Journal of Systems Science, 7(8), 953–956.
  15. [15] Chugh, R., Rani, M., and Ashish (2012), Logistic map in Noor orbit, Chaos and Complexity Letters, 6(3), 167–175.
  16. [16] Cao, J. and Chugh, R. (2019), Controlling chaos using superior feedback technique with applications in discrete traffic models, International Journal of Fuzzy Systems, 21(5), 1467-1479.
  17. [17]  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(02), 959–975.
  18. [18] Ashish and Cao, J.(2019), A novel fixed point feedback approach studying the dynamical behaviors of standard logistic map, International Journal of Bifurcation and Chaos, 29(01),1950010.
  19. [19] Peitgen, H., Jurgens, H., and Saupe, D. (2004), Chaos and Fractals, Springer: New York.
  20. [20] Sharkovsky, A.N., Maistrenko, Y.L., and Romanenko, E.Y. (1993), Difference Equations and Their Applications, Kluwer Academic Publisher: Dordrecht.
  21. [21] May, R. (1976), Simple mathematical models with very complicated dynamics, Nature, 261, 459-475.
  22. [22] Jaganathan, R. and Sinha, S. (2005), A $q$-deformed nonlinear map, Physics Letters A, 338, 277-287.
  23. [23] Patidar, V. (2006), Co-existence of regular and chaotic motions in the Gaussian map, Electronic Journal of Theoretical Physics, 3, 29-40.
  24. [24] Patidar, V. and Sud, K.K. (2009), A comparative study on the co-existing attractors in the Gaussian map and its $q$-deformed version, Communications in Nonlinear Science and Numerical Simulation, 14, 827-838.
  25. [25] Patidar, V., Purohit, G., and Sud, K.K. (2010), A numerical exploration of the dynamical behavior of nonlinear maps, Chaotic Systems: Theory and Applications, World Scientific, Singapore, 257-267.
  26. [26] Patidar, V., Purohit, G., and Sud, K.K. (2011), Dynamical Behavior of $q$-deformed Henon map, International Journal of Bifurcation and Chaos, 21, 1349-1356.
  27. [27] Shrimali, M.D. and Banerjee, S. (2013), Delayed $q$-deformed logistic map, Communications in Nonlinear Science and Numerical Simulation, 18, 3126-3133.
  28. [28] Prasad, B. and Katiyar, K. (2015), Stability and Lyapunov exponents of a $q$-deformed map, International Journal of Pure and Applied Mathematics, 104(4), 509-516.
  29. [29] Canovas, J. and Munoz-Guillermo, M. (2019), On the dynamics of the $q$-deformed logistic map, Physics Letters A, 383, 1742-1754.
  30. [30] Canovas, J. and Munoz-Guillermo, M. (2020), On the dynamics of the $q$-deformed Gaussian map, International Journal of Bifurcation and Chaos, 30(8), 2030021.
  31. [31] Wu, G.C., Cankaya, M.N., and Banerjee, S. (2020), Fractional $q$-deformed chaotic maps: A weight function approach, Chaos, 30, 121106.
  32. [32] Munoz-Guillermo, M. (2021), Image encryption using $q$-deformed logistic map, Information Sciences, 552, 352-364.
  33. [33] Mann, W.R. (1953), Mean value methods in iteration, Proceedings of American Mathematical Society, 4, 506–510.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates