Infinitely Many Coexisting Attractors and Scrolls in a Fractional-Order Discrete Neuron Map

Abstract

The neural network activation functions enable neural networks to have stronger fitting abilities and richer dynamical behaviors. In this paper, an improved fractional-order discrete tabu learning neuron (FODTLN) model map with a nonlinear periodic function as the activation function is proposed. The fixed points of the map are discussed. Then, the rich and complex dynamical behaviors of the map under different parameters and order conditions are investigated by using some common nonlinear dynamical analysis methods combined with the fractional-order approximate entropy method. Furthermore, it is found that fractional-order differential operators affect the generation of multiscrolls, and the model has infinitely many coexisting attractors obtained by changing the initial conditions. Interestingly, attractor growth and state transition are found. Finally, the map is implemented on the DSP hardware platforms to verify the realizability. The results show that the map exhibits complex and interesting dynamical behaviors. It provides a fundamental theory for the research of artificial neural networks.