A Computer Simulated Study on Applications of Fractals and Dynamical Systems

Abstract

newlineIn this dissertation, chapter 1 consists of Introduction, the statement of problems, origin of fractals and its related concepts such as self similarity, chaos, fixed point theorem. Chapter 2 and 3 covers the history of fractals and detailed Literature reviews on complex nature of fractal geometry further, various iterative methods and their applications in escape time fractals. In chapter 4 we study the effect of additive and multiplicative noise on Mandelbrot map via superior iterates and Ishikawa iterates. It includes mathematical formulation of Mann iteration and Ishikawa iteration, with the conclusion that relative superior orbit is better than Mann orbit. Chapter 5 includes the study of Tricorn and Multicorns and their behavior by using Noor orbit. Further, their properties and resulting significant structures are covered. Chapter 6 covers the study of a variant of Mandelbrot set namely composite Mandelbrot set using Noor iteration. Further, study their geometrical and analytical properties. In chapter 7, the application of Iterative functions in the field of Computer Science are explained. An effective complex key encryption and decryption model introduced using the dynamical system and fractal sets and their uses. Chapter 8 continues with the Fractal Key encryption model with most efficient key combination using multiple complex keys by the dynamical system and fractal sets. The complex key algorithm presented here is capable of defending the notorious attacks with the help of open key space and big key size. Chapter 9 Contains Implementation of multimedia encryption technique by using encryption and decryption model to encrypt and decrypt audio files, using dynamical system and fractals. Chapter 10 this chapter summarizes the thesis, discusses its findings and contributions. And also highlights outcomes of the study and paves the path for future scope of research, covers the research findings which would be useful in defining the contribution towards the future research. newline newline