Associative memories based on delayed fractional-order neural networks and application to explaining-lesson skills assessment of normal students: from the perspective of multiple $\mathit{O}(t^{-\alpha })$ stability

This paper discusses associative memories based on time-varying delayed fractional-order neural networks (DFNNs) with a type of piecewise nonlinear activation function from the perspective of multiple $\mathit{O}(t^{-\alpha })$ stability. Some sufficient conditions are gained to assure the existence of $5^n$ equilibria for $n$-neuron DFNNs with the proposed piecewise nonlinear activation functions. Additionally, the criteria ensure the existence of at least $3^n$ equilibria that are locally multiple $\mathit{O}(t^{-\alpha })$ stable. Furthermore, we apply these results to a more generic situation, revealing that DFNNs can attain $(2k+1{)}^n$ equilibria, and among them, $(k+1{)}^n$ equilibria are locally $\mathit{O}(t^{-\alpha })$ stable. Here, the parameter $k$ is highly dependent on the sinusoidal function frequency in the expanded activation functions. Such DFNNs are well-suited to synthesize high-capacity associative memories; the design process is given via singular value decomposition. Ultimately, four illustrative examples, including applying neurodynamic associative memory to the explaining-lesson skills assessment of normal students, are supplied to validate the efficacy of the results.