Stability analysis of fractional order neural networks

In this thesis, studies have been carried out the existence, uniqueness, and stabi- lity of equilibrium points of artificial neural network models that involve fractional order derivative. The thesis consists of five chapters. The first chapter provides general information about the history of artificial neural networks. In the second chapter the relationship between artificial neural networks and biological neural networks, various types of artificial neural networks, and mathematical examinations of artificial neural networks have been discussed. Furthermore, it explains the concepts of fractional derivative and integral and discusses researches carried out artificial neural network models involving fractional derivatives. The third chapter focuses on neural network models with conformable fractional derivative. In this context, the properties of comformable fractional derivative is examined. Then, the existence, uniqueness, and the exponential stability of the equilibrium point of a Hopfield-type artificial neural network are proven. Similar results are obtained for the BAM-type recurrent neural network model. Moreover, the results of these studies are supported by numerical examples. In the fourth chapter, the properties of the Caputo-Hadamard fractional derivative operator are examined. Subsequently, sufficient conditions for the existence-uniqueness and Mittag-Leffler stability of the equilibrium point of Hopfield-type artificial neural network model involving the Caputo-Hadamard fractional derivative are obtained. Numerical examples are taken into consideration and the results of this section are supported by the graphs. In the final chapter of thesis, contribution of the obtained results to the literature is explained, and possible future research using the methods employed in this thesis is discussed.