Studies On Chaos In Fractional Maps

In recent years, fractional calculus has been successfully applied to model problemsin the fields of engineering, physics, chemistry, and other sciences, such as viscoelasticmaterial, optical and thermal systems, signal processing and system identification, controland robotics, etc. Compared to the integer-order model, the significant advantage of frac-tional model is that it is more suitable to describe the material and process which have thecharacteristics of memory and hereditary.In this dissertation, there are five chapters, which are simply introduced as follows.In Chapter1, we briefly review the history of fractional calculus, and present some ba-sic notions and the corresponding properties: Riemann-Liouville derivative, Caputo deriva-tive.In Chapter2, we introduce the history of chaos, and give the definition of Li-Yorkechaos and Devaney chaos. Besides, we present the definition of the Lyapunov exponent andthe largest Lyapunov exponent. We also introduce its numerical computation method–Wolf.In Chapter3, we explore the fractional He′non map, fractional Tent map, and frac-tional Lozi map. On the one hand, we briefly review the fractional backward diference andsummation, generalized fractional backward diference and summation where the lowerbound is a positive integer, the fractional backward diference and summation in Riemann-Liouville sense, and their properties. On the other hand, through the integer He′non map,Tent map and Lozi map we show the theoretical derivation of the fractional He′non map,fractional Tent map, and fractional Lozi map respectively then discuss their non-chaoticbehaviors.In Chapter4, we further discuss the problem fractional maps. We make non-chaoticfractional maps chaotic by constructing suitable controllers. The presented control tech-nique and method has been applied to the non-chaotic fractional Tent map, fractional He′nonmap, and fractional Lozi map, in order to be chaotic via the designed controllers. The com-puter graphics are displayed to show the efciency of the designed controllers.In Chapter5, we mainly summarize the conclusions and give further studies in thefuture.