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Discrete Chaotification of a Modulated Logistic System

Ashish

Department of Mathematics, Government College Satnali, Mahendergarh, Haryana 123024, India

E-mail Address: drashishkumar108@gmail.com

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Jinde Cao

School of Mathematics, Southeast University, Nanjing, Jiangsu Province 210096, P. R. China

Yonsei Frontier Lab, Yonsei University, Seoul, Korea

E-mail Address: jdcao@seu.edu.cn

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, and

Renu Chugh

Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 124001, India

E-mail Address: chugh.r1@gmail.com

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https://doi.org/10.1142/S0218127421500656Cited by:11 (Source: Crossref)

Abstract

In the last few decades, the discrete chaotification of difference equations has gained the considerable attention of academicians and scholars due to its tremendous applications in many branches of science, such as cryptography, traffic control models, secure communications, weather forecasting and engineering. In this article, a modulated logistic system is studied and superior chaos is reported through period-doubling bifurcation, period-three orbit and Lyapunov exponent properties. It is interesting to see that the modulated system admits the superiority in chaos due to the presence of an extra degree of freedom of an ordered parameter $α$ $α$ . Analytical as well as numerical simulations are demonstrated to examine the period-doubling and Lyapunov exponent properties for some particular values of parameter $α$ $α$ . Moreover, it is speculated that the superiority in chaos may be used in chaos-based cryptography, in the future.

Keywords:

References

Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1996] Chaos: An Introduction to Dynamical Systems (Springer-Verlag, NY). Crossref, Google Scholar
Alvarez, G. & Li, S. [2006] “ Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcation and Chaos 16, 2129–2151. Link, Web of Science, Google Scholar
Andrecut, M. [1998] “ Logistic map as a random number generator,” Int. J. Mod. Phys. B 12, 101–102. Link, Web of Science, Google Scholar
Ashish , Cao, J. & Chugh, R. [2018] “ Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model,” Nonlin. Dyn. 94, 959–975. Crossref, Web of Science, Google Scholar
Ashish & Cao, J. [2019] “ A novel fixed point feedback approach studying the dynamical behaviour of standard logistic map,” Int. J. Bifurcation and Chaos 29, 1950010-1–16. Link, Web of Science, Google Scholar
Ashish , Cao, J. & Chugh, R. [2019] “ Controlling chaos using superior feedback technique with applications in discrete traffic models,” Int. J. Fuzzy Syst. 21, 1467–1479. Crossref, Web of Science, Google Scholar
Ausloos, M. & Dirickx, M. [2006] The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications (Springer-Verlag, NY). Crossref, Google Scholar
Baptista, M. S. [1998] “ Logistic map as a random number generator,” Phys. Lett. A 240, 50–54. Crossref, Web of Science, Google Scholar
Bourbakis, N. & Alexopoulos, C. [1992] “ Picture data encryption using scan patterns,” Patt. Recogn. 25, 567–581. Crossref, Web of Science, Google Scholar
Chugh, R., Rani, M. & Ashish [2012] “ Logistic map in Noor orbit,” Chaos Compl. Lett. 6, 167–175. Google Scholar
de Oliveira, L. P. L. & Sobottka, M. [2008] “ Cryptography with chaotic mixing,” Chaos Solit. Fract. 03, 466–471. Crossref, Web of Science, Google Scholar
Devaney, R. L. [1948] An Introduction to Chaotic Dynamical Systems, 2nd edition (CRC Press). Google Scholar
Devaney, R. L. [1992] A First Course in Chaotic Dynamical Systems: Theory and Experiment (CRC Press). Google Scholar
Diamond, P. [1976] “ Chaotic behaviour of systems of difference equations,” Int. J. Syst. Sci. 7, 953–956. Crossref, Web of Science, Google Scholar
Elaydi, S. N. [1999] An Introduction to Difference Equations (Springer-Verlag, NY). Crossref, Google Scholar
Feigenbaum, M. J. [1978] “ Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19, 25–52. Crossref, Web of Science, Google Scholar
Gao, H., Zhang, Y., Liang, S. & Li, D. [1978] “ A new chaotic algorithm for image encryption,” Chaos Solit. Fract. 19, 25–52. Google Scholar
Harikrishnan, K. P. & Nandakumaran, V. M. [1987] “ Bifurcation structure and Lyapunov exponent of a modulated logistic map,” Pramana-J. Phys. 29, 533–542. Crossref, Web of Science, Google Scholar
Holmgren, R. A. [1994] A First Course in Discrete Dynamical Systems (Springer-Verlag, NY). Crossref, Google Scholar
Kocarev, L. & Jakimoski, G. [2001] “ Logistic map as a block encryption algorthm,” Phys. Lett. A 289, 199–206. Crossref, Web of Science, Google Scholar
Kumar, V., Khamosh & Ashish [2020] “ An empirical approach to study the stability of generalized logistic map in superior orbit,” Adv. Math. Sci. J. 9, 8375–8384. Crossref, Google Scholar
Li, C., Lin, D., Feng, B. & Hao, F. [2018] “ Cryptanalysis of a chaotic image encryption algorithm based on information entropy,” IEEE Access 6, 75834–75842. Crossref, Web of Science, Google Scholar
Li, C., Feng, B., Li, S., Kurths, J. & Chen, G. [2019a] “ Dynamic analysis of digital chaotic maps via state-mapping networks,” IEEE Trans. Circuts Syst.-I 66, 2322–2335. Google Scholar
Li, C., Zhang, Y. & Xie, E. Y. [2019b] “ When an attacker meets a cipher-image in 2018: A year in review,” J. Inf. Secur. Appl. 48, 102361. Web of Science, Google Scholar
Lorenz, E. N. [1963] “ Deterministic non-periodic flow,” J. Atmos. Sci. 20, 130–141. Crossref, Web of Science, Google Scholar
Martelli, M. [1999] Chaos: An Introduction to Discrete Dynamical Systems and Chaos (Wiley-Interscience Publication, NY). Crossref, Google Scholar
May, R. [1976] “ Simple mathematical models with very complicated dynamics,” Nature 261, 459–475. Crossref, Web of Science, Google Scholar
Mukhamedov, F. & Mendes, J. J. J. [2007] “ On the chaotic behaviour of generalized logistic p-adic dynamical system,” J. Diff. Eqs. 243, 125–145. Crossref, Web of Science, Google Scholar
Pareek, N. K., Patidar, V. & Sud, K. K. [2003] “ Discrete chaotic cryptography using external key,” Phys. Lett. A 309, 75–82. Crossref, Web of Science, Google Scholar
Pareek, N. K., Patidar, V. & Sud, K. K. [2005] “ Cryptography using multiple one-dimensional chaotic maps,” Commun. Nonlin. Sci. Numer. Simul. 10, 715–723. Crossref, Google Scholar
Pecora, L. M. & Carroll, T. L. [1990] “ Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821. Crossref, Web of Science, Google Scholar
Poincaré, H. [1899] Les Methods Nouvells de la Mecanique Leleste (Gauthier Villars, Paris). Google Scholar
Radwan, A. G. [2013] “ On some generalized discrete logistic map,” J. Adv. Res. 4, 163–171. Crossref, Google Scholar
Robinson, C. [1995] Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (CRC Press). Google Scholar
Scharinger, J. [1998] “ Fast encryption of image data using chaotic Kolmogorov flows,” J. Electron Imag. 7, 318–325. Crossref, Web of Science, Google Scholar
Sharkovsky, A. N., Maistrenko, Y. L. & Romanenko, E. Y. [1993] Difference Equations and Their Applications (Kluwer Academic Publishers, Netherlands). Crossref, Google Scholar
Singh, N. & Sinha, A. [2010] “ Chaos-based secure communication system using logistic map,” Opt. Lasers Eng. 48, 398–404. Crossref, Web of Science, Google Scholar
Smaoui, N. & Kanso, A. [2009] “ Cryptography with chaos and shadowing,” Chaos Solit. Fract. 42, 2312–2321. Crossref, Web of Science, Google Scholar
Strogatz, S. H. [1994] Nonlinear Dynamics and Chaos (Perseus Books Publishing, NY). Google Scholar
Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamics and Chaos (Springer-Verlag, NY). Crossref, Google Scholar
Yen, J. C. & Guo, J. I. [1999] “ A new image encryption algorithm and its VLSI architecture,” Proc. IEEE Workshop Sign. Proc. Syst., pp. 430–437. Google Scholar
Zhang, X. & Cao, Y. [2014] “ A novel chaotic map and an improved chaos based image encryption scheme,” Sci. World J. 2014, 713541-1–8. Google Scholar
Zhu, H., Qi, W., Ge, J. & Liu, Y. [2018] “ Analyzing Devaney chaos of a sine-cosine compound function system,” Int. J. Bifurcation and Chaos 28, 1850176-1–13. Link, Web of Science, Google Scholar
Zhu, H., Zhao, Y. & Song, Y. [2019] “ 2D logistic-modulated-sine-coupling-logistic chaotic map for image encryption,” IEEE Access 7, 14081–14098. Crossref, Web of Science, Google Scholar