On investigation of the complex dynamics of crisp and fuzzy fractals using iterative techniques

The complex dynamics of crisp and fuzzy fractals within the dynamical and parameter planes encompass self-similarity across different scales, intricate patterns, and non-integer dimensions. They exhibit structured characteristics defined by iterative methods, maintain invariance under scaling, and often demonstrate non-Euclidean geometry. Julia and Mandelbrot’s sets exemplify one fractal type, with iterative methods fundamental to their creation. In this dissertation, we delve into the dynamics of two types of fractals: crisp and fuzzy fractals. Our exploration begins with an investigation into the dynamics of crisp fractals through a complex-valued mapping C ∋ z → z^p − qz^2 + rz + sin c^w, where p ∈ N \ {1}, w ∈ [1, ∞), r, c ∈ C \ {0}, and q ∈ C \ {1}. Using the Mann, Picard-Mann, and four-step iterative methods with s-convexity, we prove an escape criterion, formulate algorithms, implement these algorithms, and generate novel patterns of crisp Julia sets, crisp Mandelbrot sets, and crisp biomorphs. We identified shortcomings in the literature regarding the proof of escape criteria using iterative methods with s-convexity. One of the aims of this dissertation is to address and rectify these flaws comprehensively. We prove the compactness of the crisp Mandelbrot sets as well as the crisp Julia set’s symmetry around the real axis generated using the Mann iterative method with s-convexity under specific conditions. Additionally, we conduct numerical experiments to explore execution time and the average iteration count, enhancing our understanding of fractal dynamics and computational efficiency in fractal generation and size analysis. The second part of this dissertation introduces the groundbreaking concept of fuzzy fractals (fuzzy Julia sets, fuzzy Juliabar sets, and fuzzy biomorphs) within the dynamical plane, advancing the foundational principles established by crisp fractals using a complex-valuedmappings C ∋ z → z^p + c and C ∋ z → z¯^p + c, where p ∈ N \ {1} and c ∈ C alongside iterative processes that capture the gradual divergence of orbits towards infinity using nuanced membership functions. We present a detailed explanation of these sets, defining their membership functions with examples. Membership functions measure the degree of relationship of points with these sets via iterative processes, even as their orbits diverge towards infinity. Through meticulously formulated algorithms, we generate two-dimensional and three-dimensional fuzzy fractals that transcend the limitations of crisp representations, revealing intricate patterns. Furthermore, this work establishes the axial and rotational symmetries of fuzzy Julia and Juliabar sets, characterized by a rotational symmetry of an angle 2π/p radians. This dissertation advances fractal theory with detailed explanations, examples, rigorous proofs, and computational analyses. It opens new avenues in fuzzy fractal geometry, enhancing our understanding of fractal patterns.