| Summary: | 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.
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