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The coexistence of multiple stable equilibria in recurrent neural networks is an important dynamic characteristic for associative memory and other applications. In this paper, the existence and local Mittag-Leffler stability of multiple equilibria are investigated for a class of fractional-order recurrent neural networks with discontinuous and nonmonotonic activation functions. By using Brouwer s fixed point theory, several conditions are established to ensure the existence of 5 n equilibria, in which all the components of 4 n equilibria are located in the continuous intervals of the activation functions. and some of the components of 5 n − 4 n equilibria are located at some discontinuous points of the activation functions. The introduction of discontinuous activation functions makes the neural networks have more equilibria than those with continuous activation functions. Furthermore, some criteria are proposed to ensure local Mittag-Leffler stability of 3 n equilibria. The introduction of nonmonotonic activation functions makes the neural networks have more stable equilibria than those with monotonic activation functions. Two examples are given to illustrate the effectiveness of the results. INDEX TERMS Fractional-order recurrent neural network, discontinuous and nonmonotonic activation function, equilibrium point, local Mittag-Leffler stability.

The stabilization of first‐order delayed (FOD) unstable processes with proportional–integral–derivative (PID) controllers is considered, and all the feasible PID controllers are determined. Different from the existing results which are based on the D‐partition technique and partitioning complex's real and imaginary parts, a novel procedure enlightened by the τ decomposition method is proposed to characterize the space of controller parameters. Generally speaking, the parameter space is mainly divided into four regions: delay independent stable region, delay independent unstable region, delay interval dependent stable region, and delay interval dependent unstable region. The depiction of the space partition boundaries and the determination of stable intervals become the main problems. In our work, these boundaries are related to the existence of the purely imaginary roots (PIRs) of the systems' characteristic equation, and the determination of stable intervals refers to the number and value of the PIRs. Thus, the key points are the number and calculation of PIRs. According to our discussion, analytical results on both topics are obtained in explicit form. Finally, vivid numerical simulations are given to illustrate the results.

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