Stabilization of Chaotic Fractional Discrete Neural Networks Based on the Caputo -h- Difference Operator

Abstract

The objective of this thesis is to present a general study of discrete fractional-order dynamical systems, with a particular focus on chaotic fractional discrete neural networks, stability theory, and chaotic behavior. More specifically, we are interested in fractional difference systems, including Caputo and h-fractional operators, where memory effects play a fundamental role in the system dynamics.

The study investigates both linear and nonlinear discrete fractional models and dis- cusses several stability concepts, including Mittag–Leffler stability, which is widely used

in the analysis of fractional systems. In addition, chaotic dynamics are examined through

mathematical tools such as Lyapunov exponents, bifurcation analysis, and fractal dimen- sions, providing insight into the complex behavior of nonlinear discrete systems.

Furthermore, fractional-order discrete neural networks are considered as an important application of these systems. Neural networks have demonstrated remarkable capabilities in various tasks, including pattern recognition, image classification, natural language processing, and regression analysis. They have found applications in diverse fields such as finance, healthcare, marketing, robotics, and autonomous vehicles, among others.