Nonlinear Dynamics

, Volume 78, Issue 2, pp 771–778 | Cite as

Analysis of a new simple one dimensional chaotic map

  • Orcan Alpar
Original Paper


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.


Chaotic 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.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Business Management, Discipline of Quantitative Methods, Institute of Social SciencesIstanbul UniversityIstanbulTurkey

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