Stability of Delay Hopfield Neural Networks with Generalized Riemann-Liouville Type Fractional Derivative

Area of Science:

• Computational Neuroscience
• Dynamical Systems Theory
• Fractional Calculus

Background:

• Hopfield neural networks are fundamental models in computational neuroscience.
• Previous models often simplified delay dynamics and derivative types.
• Incorporating fractional derivatives and complex delays presents significant analytical challenges.

Purpose of the Study:

• To investigate the stability and convergence of a generalized Hopfield neural network.
• To analyze the impact of time-varying delays and a specific fractional derivative on network dynamics.
• To develop novel analytical methods for fractional-order systems with delays.

Main Methods:

• Utilized a generalized Riemann-Liouville fractional derivative (GRLFD) with Sonine kernels.
• Applied Lyapunov-type convex functions and the Razumikhin method for stability analysis.
• Established comparison results for Lyapunov functions tailored to the GRLFD.

Main Results:

• Proved an inequality for Lyapunov functions involving the GRLFD.
• Established exponential bounds for solutions, excluding the initial time.
• Demonstrated the convergence of all solutions to a constant equilibrium at infinity.

Conclusions:

• The study provides a rigorous framework for analyzing complex fractional-order neural networks.
• Theoretical results offer insights into the stability and long-term behavior of systems with delays and fractional dynamics.
• The findings are illustrated with an example, highlighting their practical relevance.