Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays
This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the qu...
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MDPI AG
2022-06-01
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| author | Wenjun Dong Yujiao Huang Tingan Chen Xinggang Fan Haixia Long |
| author_facet | Wenjun Dong Yujiao Huang Tingan Chen Xinggang Fan Haixia Long |
| author_sort | Wenjun Dong |
| collection | DOAJ |
| description | This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>3</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>3</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> equilibrium points. According to the Halanay inequality, the conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example. |
| first_indexed | 2024-03-09T04:02:45Z |
| format | Article |
| id | doaj.art-9ffa9a9b21664ab085471a94539c4711 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:02:45Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9ffa9a9b21664ab085471a94539c47112023-12-03T14:11:27ZengMDPI AGMathematics2227-73902022-06-011013215710.3390/math10132157Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time DelaysWenjun Dong0Yujiao Huang1Tingan Chen2Xinggang Fan3Haixia Long4College of Zhijiang, Zhejiang University of Technology, Shaoxing 312030, ChinaCollege of Zhijiang, Zhejiang University of Technology, Shaoxing 312030, ChinaCollege of Zhijiang, Zhejiang University of Technology, Shaoxing 312030, ChinaCollege of Zhijiang, Zhejiang University of Technology, Shaoxing 312030, ChinaCollege of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310058, ChinaThis study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>3</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>3</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> equilibrium points. According to the Halanay inequality, the conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mrow><mn>4</mn><mi>n</mi></mrow></msup></semantics></math></inline-formula> local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example.https://www.mdpi.com/2227-7390/10/13/2157quaternion-valued neural networklocal Lagrange exponential stabilitymultiple equilibrium pointstime delay |
| spellingShingle | Wenjun Dong Yujiao Huang Tingan Chen Xinggang Fan Haixia Long Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays Mathematics quaternion-valued neural network local Lagrange exponential stability multiple equilibrium points time delay |
| title | Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays |
| title_full | Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays |
| title_fullStr | Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays |
| title_full_unstemmed | Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays |
| title_short | Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays |
| title_sort | local lagrange exponential stability analysis of quaternion valued neural networks with time delays |
| topic | quaternion-valued neural network local Lagrange exponential stability multiple equilibrium points time delay |
| url | https://www.mdpi.com/2227-7390/10/13/2157 |
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