Chaotic SystemsofFractionalOrder

Abstract

This thesis investigates the bifurcation and stability properties of discrete systems induced by fractional-order continuous chaotic finance systems and Arneodo's chaotic system. The research is structured into four chapters, each focusing on different aspects related to fractional calculus, chaos theory, discretization methods, and the main results obtained from the analysis. Chapter 1 and Chapter 2 provide a preliminary introduction to fractional calculus, presenting the necessary mathematical tools and concepts for understanding the fractional order systems. Additionally, it discusses the fundamentals of chaos theory, emphasizing the significance of chaos. In Chapter 3, various discretization methods are examined to transform the continuous fractional-order chaotic systems into discrete counterparts. The discretization techniques are thoroughly analyzed, considering their impact on the system dynamics and preserving the essential chaotic features during the discretization process. Chapter 4 presents the main results of the thesis. Firstly, it investigates the bifurcation and stability properties of a new discrete system induced from a fractional-order continuous chaotic finance system. Through numerical simulations and mathematical analysis, the chapter reveals the existence of bifurcation points and characterizes their influence on the system's stability. Secondly, a similar analysis is conducted on a discrete system induced from the chaotic Arneodo's system, exploring its bifurcation patterns and stability properties.