Multi-stability analysis of fractional-order quaternion-valued neural networks with time delay

This paper addresses the problem of multi-stability analysis for fractional-order quaternion-valued neural networks (QVNNs) with time delay. Based on the geometrical properties of activation functions and intermediate value theorem, some conditions are derived for the existence of at least $(2{\mathcal{K}}_p^R+1{)}^n,(2{\mathcal{K}}_p^I+1{)}^n,(2{\mathcal{K}}_p^J+1{)}^n,(2{\mathcal{K}}_p^K+1{)}^n$ equilibrium points, in which $[({\mathcal{K}}_p^R+1){]}^n,[({\mathcal{K}}_p^I+1){]}^n,[({\mathcal{K}}_p^J+1){]}^n,[({\mathcal{K}}_p^K+1){]}^n$ of them are uniformly stable while the other equilibrium points become unstable. Thus the developed results show that the QVNNs can have more generalized properties than the real-valued neural networks (RVNNs) or complex-valued neural networks (CVNNs). Finally, two simulation results are given to illustrate the effectiveness and validity of our obtained theoretical results.