PaperNo Access

A Novel Fixed Point Feedback Approach Studying the Dynamical Behaviors of Standard Logistic Map

Ashish

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

E-mail Address: 101101515@seu.edu.cn

Search for more papers by this author

and

Jinde Cao

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

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

Search for more papers by this author

https://doi.org/10.1142/S021812741950010XCited by:17 (Source: Crossref)

Abstract

The standard logistic map is one of the oldest and simplest systems which has found a celebrated place in the dynamical systems and in different vital applications of science like image encryption in cryptography, secure communications and traffic control models. Generally, the dynamical systems are characterized by one or more control parameters that determine the dynamical behaviors of the system. Traditionally, the discrete logistic map allows only one parameter $λ \in [0, 4]$ to determine its complete behavior. This study takes one step forward, using the superior fixed point iterative technique to study the dynamical properties of the discrete logistic map. The proposed technique provides an extra degree of freedom on control parameters that renders superior dynamical properties and may increase the performance of many applications. Analytical analysis as well as numerical simulations are presented to show the effectiveness, flexibility and efficiency of new method.

Keywords:

References

Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1996] Chaos: An Introduction to Dynamical Systems (Springer-Verlag, NY). 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
Ausloos, M. & Dirickx, M. [2006] The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications (Springer-Verlag, NY). Google Scholar
Baptista, M. S. [1998] “ Logistic map as a random number generator,” Phys. Lett. A 240, 50–54. Web of Science, Google Scholar
Block, L. S. & Coppel, W. A. [1992] Dynamics in One Dimension (Springer-Verlag, NY). Google Scholar
Chen, G. & Huang, Y. [2011] Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations (Morgan and Claypool Publishers, USA). Google Scholar
Chugh, R., Rani, M. & Ashish [2012a] “ Logistic map in Noor orbit,” Chaos Compl. Lett. 6, 167–175. Google Scholar
Chugh, R., Rani, M. & Ashish [2012b] “ On the convergence of logistic map in Noor orbit,” Int. J. Comp. Appl. 43, 1–4. Google Scholar
Crownover, R. M. [1995] Introduction to Fractals and Chaos (Jones and Barlett Publishers, Burlington). Google Scholar
de Oliveira, L. P. L. & Sobottka, M. [2008] “ Cryptography with chaotic mixing,” Chaos Solit. Fract. 3, 466–471. Web of Science, Google Scholar
Devaney, R. L. [1948] An Introduction to Chaotic Dynamical Systems, 2nd edition (Addison-Wesley). Google Scholar
Devaney, R. L. [1992] A First Course in Chaotic Dynamical Systems: Theory and Experiment (Addison-Wesley). Google Scholar
Diamond, P. [1976] “ Chaotic behaviour of systems of difference equations,” Int. J. Syst. Sci. 7, 953– 956. Web of Science, Google Scholar
Feigenbaum, M. J. [1978] “ Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19, 25–52. Web of Science, Google Scholar
Hogan, S. J., Champneys, A. R., Krauskopf, B., di Bernardo, M., Wilson, R. E., Osinga, H. M. & Homer, M. E. (eds.) [2003] Nonlinear Dynamics and Chaos: Where Do We Go from Here? (IOP Publishing, Ltd.). Google Scholar
Holmgren, R. A. [1994] A First Course in Discrete Dynamical Systems (Springer-Verlag, NY). Google Scholar
Lloyd, A. L. [1995] “ The coupled logistic map: A simple model for the effects of spatial heterogeneity on population dynamics,” J. Theor. Biol. 173, 217–230. Web of Science, Google Scholar
Lorenz, E. N. [1963] “ Deterministic non-periodic flow,” J. Atmos. Sci. 20, 130–141. Web of Science, Google Scholar
Malek, K. & Gobal, F. [2000] “ Application of chaotic logistic map for the interpretation of anion-insertion in poly-ortho-aminophenol films,” Synth. Met. 113, 167–171. Web of Science, Google Scholar
Mann, W. R. [1953] “ Mean value methods in iteration,” Proc. Amer. Math. Soc. 4, 506–510. Web of Science, Google Scholar
Martin, J. S. & Porter, M. A. [2014] “ Convergence time towards periodic orbits in discrete dynamical systems,” PLoS One 9, 1–9. Web of Science, Google Scholar
May, R. [1976] “ Simple mathematical models with very complicated dynamics,” Nature 261, 459–475. Web of Science, Google Scholar
Medina, R. V., Mendez, A. D., Rio-Correa, J. L. & Hernandez, J. L. [2009] “ Design of chaotic analog noise generators with logistic map and MOS QT circuits,” Chaos Solit. Fract. 40, 1779–1793. Web of Science, Google Scholar
Molina, C., Sampson, N., Fitzgerald, W. J. & Niranjan, M. [1996] “ Geometrical techniques for finding the embedding dimension of time series,” Proc. IEEE Signal Processing Society Workshop, pp. 161–169. Google Scholar
Peitgen, H., Jurgens, H. & Saupe, D. [2004] Chaos and Fractals (Springer-Verlag, NY). Google Scholar
Rani, M. & Agarwal, R. [2009] “ A new experimental approach to study the stability of logistic map,” Chaos Solit. Fract. 4, 2062–2066. Web of Science, Google Scholar
Robinson, C. [1995] Dynamical Systems: Stabilily, Symbolic Dynamics, and Chaos (CRC Press). Google Scholar
Salarieh, H. & Alasty, A. [2008] “ Stabilizing unstable fixed points of chaotic maps via minimum entropy control,” Chaos Solit. Fract. 3, 763–769. Web of Science, Google Scholar
Sharkovsky, A. N., Maistrenko, Y. L. & Romanenko, E. Y. [1993] Difference Equations and Their Applications (Kluwer Academic Publisher). Google Scholar
Singh, N. & Sinha, A. [2010] “ Chaos-based secure communication system using logistic map,” Opt. Lasers Eng. 48, 398–404. Web of Science, Google Scholar
Song, N. & Meng, J. [1996] “ Research on logistic mapping and synchronization,” Proc. IEEE Intell. Contr. Autom. 1, 987–991. Google Scholar
Wackerbauer, R., Witt, A., Atmanspacher, H., Kurths, J. & Scheingraber, H. [1994] “ A comparative classification of complexity measures,” Chaos Solit. Fract. 1, 133–173. Web of Science, Google Scholar
Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamics and Chaos (Springer-Verlag, NY). Google Scholar