Complexity and attractor–repeller applications in fractals visualization using T-iteration

In this paper, we use a recently developed fixed-point iteration method called the T-iterative scheme to construct and analyze fractal structures. This method provides a robust and efficient approach to analyze the nonlinear dynamics of complex polynomials. We examine the influence of parameter variations on the emergence and complexity of resultant fractals by integrating quantitative metrics with visual fractal patterns. Our results show that even small changes to these parameters lead to structures that are significantly different and complicated. To quantify these observations, we provide two numerical metrics. The first one, the Complexity Index (CmI), shows how complex the generated fractals are. The second metric, the Attractor and Repeller Percentage (ARP), shows how many of the starting points move toward stable fixed points and how many move away from them. We demonstrate how changes in various parameters directly impact dynamical stability, morphological diversity, and overall fractal complexity by examining CmI and ARP concurrently.