On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia sets

Numerical Algorithms - Trang 1-26 - 2023
Sudesh Kumari1, Krzysztof Gdawiec2, Ashish Nandal3, Naresh Kumar4, Renu Chugh5
1Government College for Girls Sector 14, Gurugram, India
2Institute of Computer Science, University of Silesia, Sosnowiec, Poland
3Government College Baund Kalan, Charkhidadri, India
4Department of Mathematics, Guru Jambheshwar University of Science & Technology, Hisar, India
5Department of Mathematics, Gurugram University, Gurugram, India

Tóm tắt

In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form $$T(z) = z^{n} + mz + r$$ , where $$n \ge 2$$ and $$m, r \in \mathbb {C}$$ , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear.

Tài liệu tham khảo

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