Fractional Discrete Chaos

In the nineteenth-century, fractional calculus had its origin in extending differentiation and integration operators from the integer-order case to the fractional-order case. Discrete fractional calculus has recently become an important research topic, useful in various science and engineering applications. The first definition of the fractional-order discrete-time/difference operator was introduced in 1974 by Diaz and Osler, where such operator was derived by discretizing the fractional-order continuous-time operator. Successfully, several types of fractional-order difference operators have then been proposed and introduced through further generalizing numerous classical operators, motivating several researchers to publish extensively on a new class of systems, viz the nonlinear fractional-order discrete-time systems (or simply, the fractional-order maps), and their chaotic behaviors. This discovery of chaos in such maps, has led to novel control methods for effectively stabilizing their chaotic dynamics.

The aims of this book are as follows:

• Presenting the recent developments, trends, research solutions, applications and open problems related to fractional-order chaotic maps;
• Illustrating many interdisciplinary applications, like modulization, control, circuits, security and encryption;
• Including all theories associated with chaos, control and synchronization of discrete-time systems;
• Providing a useful reference on the topic of fractional-order chaotic maps and their applications.