Construction and Application of New Classes of Higher-order Chaotic Polynomial Maps

Abstract

Dynamical systems find various applications in diverse fields, including Computer sci ence, Mathematics, Engineering, Physics, Chaos theory, Population dynamics, Eco nomics, and Finance. This research work explores the new classes of chaotic polynomial maps and their applications. Due to the high sensitivity of these classes of chaotic poly nomial maps, these maps are utilized to generate new schemes of PRNG. This thesis is divided into three chapters. In the first chapter, we explore the dynamical system, orbits, their types, fixed points, and two theorems related to the repelling and attracting of the fixed point. We also provide a graphical analysis of the orbits to check the attracting and repelling of the orbits. Next, we explore the bifurcation theory, period-doubling bifurcation, and orbit diagram. Finally, we give the famous result of the Li-Yorke theorem, which states that if any one-dimensional continuous function has an orbit of period three, then it has orbits of any other positive integer order. In the next chapter, we review the two classes of chaotic polynomial maps based on the Li-Yorke Theorem [1] and give the theoretical proof of these classes of maps. Next, we ob serve the dynamic behavior of one map from each class using bifurcation and Lyapunov exponent plots and observe that these maps are chaotic in the given chaotic interval. Finally, we study the PRNG scheme based on these classes of maps. In the final chapter of this thesis, we constructed new classes of chaotic polynomial maps, presented the theoretical proof of these classes of chaotic polynomial maps, and proved that these maps are chaotic in the given chaotic interval by using the famous LI-Yorke result. We analyzed the dynamic behavior of these classes of chaotic polynomial maps by choosing one specific map from each class. We also constructed a new PRNG scheme with good randomness and uniformity.