Finite-time multistability of a multidirectional associative memory neural network with multiple fractional orders based on a generalized Gronwall inequality

This paper addressed the finite-time multistability of a Caputo fractional order multidirectional associative memory neural network (FMAMNN) with multiple orders, where the fractional orders are not limited to 0 to 1. There are three main findings. Firstly, by using Brouwer fixed point theorem, the existence conditions of multiple equilibria of FMAMNN with Gaussian-wavelet-type activation functions were obtained, and the number of equilibria is (2+s^2)^l , where the exponent l is determined by the number of neurons, and s is the number of segments in the middle of Gaussian-wavelet-type activation function. Another important contribution of this article is the generalization of the Gronwall inequality. By utilizing the multivariable Mittag-Leffler function, a more general form of the Gronwall inequality was obtained, which can be applied to systems with many different fractional derivatives. Lastly, based on the generalized Gronwall inequality, and using the Laplace transform, inverse Laplace transform, some conditions of finite-time multistability of FMAMNN were found, which depend on all fractional orders. Numerical calculations show that compared to the existing finite-time stability conditions of neural networks, the conditions based on the generalized Gronwall inequality have less conservatism. An example was given to show the validity of theoretical results.