Chaos, Fractals and Bifurcation in Real Dynamics of Two-Parameter Family Associated to Logarithmic Functions

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DOI: https://doi.org/10.28924/2291-8639-23-2025-13
Published: Jan 14, 2025
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Shaurya Pratap Singh, Mohammad Sajid, Chaos, Fractals and Bifurcation in Real Dynamics of Two-Parameter Family Associated to Logarithmic Functions, Int. J. Anal. Appl., 23 (2025), 13.

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Shaurya Pratap Singh, Mohammad Sajid

Abstract

The purpose of this article is to provide bifurcation diagrams and observe chaotic behaviour in the real dynamics of two-parameter family of function \(\Phi(x)=x+(1-\lambda x)\ln(ax): x>0, \lambda>0, a>0\). We consider here parameter a is a positive and continuous real parameter while λ is positive but a discrete real parameter. The dynamical properties of this nonlinear system family analyze numerically as well as graphically by using fixed point iterative method. Bifurcation diagrams for the real dynamics of the function are plotted by varying the values of the parameter which are fractals in nature. Also, we show that chaos exists in the dynamics of the function by looking at period-doubling in the bifurcation diagram. Further, chaotic behaviour studies by simulation of the positive Lyapunov exponents which observe by varying the parameters similar to the bifurcation diagrams.

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