Generation of Fractal Attractor for Controlled Metric Based Dynamical Systems

Mandelbrot initiated the term “Fractal” in 1975, and it has since gained popularity among mathematicians and physicists alike. The mathematical properties of fractals are available and applied in the chaotic structures of various systems, which are generally experienced in science and technology. The iterated function system (IFS) evolved as a practical application of the theory of discrete dynamical systems and is a valuable tool to generate fractal attractors. In this context, the Hutchinson-Barnsley (HB) theory is generalized to construct fractal sets using an IFS of contractions on a controlled metric space (CMS). The HB theorem of IFS is proved in a Hausdorff controlled metric space (HCMS), and it is also guaranteed that the HB operator has merely a single fixed point in a Hausdorff controlled metric space, known as a controlled fractal. This study also links the extended rectangular b-metric space (ERbMS) and the controlled metric space to create a new metric space, the controlled extended rectangular b-metric space (CERbMS), by incorporating the control factor as a function in the rectangular inequality. In addition, the fixed point theorem is also proved with specific conditions for contractions in the proposed metric space CERbMS and illustrated with an example. Finally, the structure of IFS is defined in CERbMS to construct the HB theory for generating the controlled extended rectangular b-fractals.