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

S. Kathiresan; Ardak Kashkynbayev; K. Janani; R. Rakkiyappan; S. Kathiresan; Ardak Kashkynbayev; K. Janani; R. Rakkiyappan

doi:10.3934/math.2022199

AIMS Mathematics

2022, Volume 7, Issue 3: 3603-3629. doi: 10.3934/math.2022199

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

1.
Department of Mathematics, Rathinam College of Arts and Science, Coimbatore-641021, Tamilnadu, India
2.
Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan
3.
Department of Mathematics, Bharathiar University, Coimbatore-641046, Tamilnadu, India

Received: 01 October 2021 Accepted: 23 November 2021 Published: 06 December 2021
MSC : 34D05, 34D06, 34D20, 34D23

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.
- multiple stability,
- fractional-order,
- caputo fractional derivative,
- quaternion-valued neural networks,
- time delay
Citation: S. Kathiresan, Ardak Kashkynbayev, K. Janani, R. Rakkiyappan. Multi-stability analysis of fractional-order quaternion-valued neural networks with time delay[J]. AIMS Mathematics, 2022, 7(3): 3603-3629. doi: 10.3934/math.2022199

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Abstract

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.

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