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Multistability of Quaternion-Valued Recurrent Neural Networks with Discontinuous Nonmonotonic Piecewise Nonlinear Activation Functions

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Abstract

In this article, the coexistence and dynamical behaviors of multiple equilibrium points for quaternion-valued neural networks (QVNNs) are investigated, whose activation functions are discontinuous and nonmonotonic piecewise nonlinear. According to the Hamilton rules, the QVNNs can be divided into four real-valued parts. By utilizing the Brouwer’s Fixed Point Theorem and property of strictly diagonally dominant matrices, some sufficient conditions are derived to ensure that the QVNNs have at least \(5^{4n}\) equilibrium points, \(3^{4n}\) of them are locally exponentially stable, and the others are unstable. It is shown that the number of stable equilibria in QVNNs is more than that in the real-valued ones. Finally, a numerical simulation is presented to clarify the theoretical analysis is valid.

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Acknowledgements

The research is supported by grants from the National Natural Science Foundation of China (Nos. 62172188, 52072130) and the Natural Science Foundation of Guangdong Province in China (No. 2021A 1515011753).

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Authors and Affiliations

  1. College of Information Science and Technology, Jinan University, Guangzhou, 510632, China

    Weihao Du, Jianglian Xiang & Manchun Tan

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  1. Weihao Du
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  2. Jianglian Xiang
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  3. Manchun Tan
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Contributions

WD: Writing—Original Draft, Formal analysis. JX: Visualization. MT: Writing—Review & Editing.

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Correspondence to Manchun Tan.

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Du, W., Xiang, J. & Tan, M. Multistability of Quaternion-Valued Recurrent Neural Networks with Discontinuous Nonmonotonic Piecewise Nonlinear Activation Functions. Neural Process Lett 55, 5855–5884 (2023). https://doi.org/10.1007/s11063-022-11116-w

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