Analysis of a new simple one dimensional chaotic map
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In this paper, a new one-dimensional map is introduced, which exhibits chaotic behavior in small interval of real numbers. It is discovered that a very simple fraction in a square root with one variable and two parameters can lead to a period-doubling bifurcations. Given the nonlinear dynamics of one-dimensional chaotic maps, it is usually seen that chaos arises when the parameter raises up to a value, however in our map, which seems reverse, it arises when the related parameter decreases and approaches to a constant value. Since proposing a new map entails solid foundations, the analysis is originated with linear stability analysis of the new map, finding fixed points. Additionally, the nonlinear dynamics analysis of the new map also includes cobweb plot, bifurcation diagram, and Lyapunov analysis to realize further dynamics. This research is mainly consisting of real numbers, therefore imaginary parts of the simulations are omitted. For the numerical analysis, parameters are assigned to given values, yet a generalized version of the map is also introduced.
KeywordsChaotic map One-dimensional Bifurcation Cobweb plot Lyapunov exponent
Researchers are kindly requested to cite this map, when needed, as “Bülban Map” or “Bülban’s Map”, in ever remembrance of the name of author’s mother.
- 3.Strogatz, S.: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering. Perseus Books, New York (1994)Google Scholar
- 5.Nusse, H., Yorke, J.: Dynamics: Numerical Explorations. Applied Mathematical Sciences, 2nd edn. Springer-Verlag, Berlin (1998)Google Scholar
- 6.Lozi, R.: Un attracteur étrange du type attracteur de Hénon. J. de Phys. Ser. C 39(5), 9–10 (1978)Google Scholar
- 12.Ou, W., Lai, X., Wu, M., Cao, W.: “Design and implementation of a new third order chaotic system”, In: 25th Chinese Control and Decision Conference (CCDC), IEEE., (2013)Google Scholar
- 14.Gao, Z. Z.: A new chaotic system and analysis of its properties. J. Sichuan Univ. Sci. & Eng. (Natural Science Edition), vol. 2, (2011)Google Scholar
- 16.Zhang, J., Tang, W.: A novel bounded 4D chaotic system. Nonlinear Dyn. 67(4), 2455–2465 (2012) Google Scholar
- 17.Liang, Z. C., Zhonglin, W.: Design and realization of a new chaotic system. In: Sensor Network Security Technology and Privacy Communication System (SNS & PCS), 2013 International Conference on, IEEE, (2013)Google Scholar
- 18.Ye, Z., Deng, C.: Adaptive synchronization to a general non-autonomous chaotic system and its applications. Nonlinear Anal.: Real World Appl. 13(2), 840–849 (2012)Google Scholar
- 23.Peters, E.: Chaos and order in the capital markets: A new view of cycles, prices, and market volatility, 2nd edn. John Wiley & Sons Inc., New York (1996)Google Scholar
- 25.Paul, A., Das, N., Prusty, A. K.: An advanced gray image encryption scheme by using discrete logarithm with logistic and HEH64 chaotic functions. In: In Advance Computing Conference (IACC), 2013 IEEE 3rd International IEEE, (2013)Google Scholar