Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays

Jiang-Wei Ke; Jin-E Zhang; Ji-Xiang Zhang; Jiang-Wei Ke; Jin-E Zhang; Ji-Xiang Zhang

doi:10.3934/math.2024220

AIMS Mathematics

2024, Volume 9, Issue 2: 4562-4586. doi: 10.3934/math.2024220

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Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received: 28 November 2023 Revised: 27 December 2023 Accepted: 05 January 2024 Published: 18 January 2024
MSC : 92B20, 93D05

This paper explores multimode function multistability of Cohen-Grossberg neural networks (CGNNs) with Gaussian activation functions and mixed time delays. We start by using the geometrical properties of Gaussian functions. The state space is partitioned into $ 3^\mu $ subspaces, where $ 0\le \mu\le n $. Moreover, through the utilization of Brouwer's fixed point theorem and contraction mapping, some sufficient conditions are acquired to ensure the existence of precisely $ 3^\mu $ equilibria for $ n $-dimensional CGNNs. Meanwhile, there are $ 2^\mu $ and $ 3^\mu-2^\mu $ multimode function stable and unstable equilibrium points, respectively. Ultimately, two illustrative examples are provided to confirm the efficacy of theoretical results.
- Cohen-Grossberg neural networks,
- multistability,
- Gaussian activation functions,
- mixed time delays,
- multimode function stability
Citation: Jiang-Wei Ke, Jin-E Zhang, Ji-Xiang Zhang. Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays[J]. AIMS Mathematics, 2024, 9(2): 4562-4586. doi: 10.3934/math.2024220

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Abstract

This paper explores multimode function multistability of Cohen-Grossberg neural networks (CGNNs) with Gaussian activation functions and mixed time delays. We start by using the geometrical properties of Gaussian functions. The state space is partitioned into $ 3^\mu $ subspaces, where $ 0\le \mu\le n $. Moreover, through the utilization of Brouwer's fixed point theorem and contraction mapping, some sufficient conditions are acquired to ensure the existence of precisely $ 3^\mu $ equilibria for $ n $-dimensional CGNNs. Meanwhile, there are $ 2^\mu $ and $ 3^\mu-2^\mu $ multimode function stable and unstable equilibrium points, respectively. Ultimately, two illustrative examples are provided to confirm the efficacy of theoretical results.

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