Dynamical properties of a novel one dimensional chaotic map

Amit Kumar; Jehad Alzabut; Sudesh Kumari; Mamta Rani; Renu Chugh; Amit Kumar; Jehad Alzabut; Sudesh Kumari; Mamta Rani; Renu Chugh

doi:10.3934/mbe.2022115

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 3: 2489-2505. doi: 10.3934/mbe.2022115

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Dynamical properties of a novel one dimensional chaotic map

1.
Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India
2.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Turkey
4.
Department of Mathematics, Government College for Girls Sector 14, Gurugram 122001, India
5.
Department of Computer Science, Central University of Rajasthan, Ajmer 305801, India

Academic Editor：Maria Luz Gandarias

Received: 09 November 2021 Revised: 24 December 2021 Accepted: 31 December 2021 Published: 07 January 2022

In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu > 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.
- nonlinear dynamics,
- chaotic map,
- stability,
- maximal lyapunov exponent,
- entropy
Citation: Amit Kumar, Jehad Alzabut, Sudesh Kumari, Mamta Rani, Renu Chugh. Dynamical properties of a novel one dimensional chaotic map[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2489-2505. doi: 10.3934/mbe.2022115

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Abstract

In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu > 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.

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