Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture

Area of Science:

• Computational neuroscience
• Dynamical systems theory
• Artificial neural networks

Background:

• Hopfield-type neural networks are crucial for associative memory and pattern recognition.
• Understanding the stability and bifurcations of these networks is essential for predicting their behavior.
• Discrete-time models with delays introduce complex dynamics not present in continuous-time counterparts.

Purpose of the Study:

• To analyze the stability domain of the null solution in a discrete-time-delayed Hopfield-type neural network with ring architecture.
• To identify critical parameter values leading to bifurcations at the origin.
• To investigate the occurrence of complex dynamics and chaotic behavior.

Main Methods:

• Center manifold theorem
• Normal form theory
• Analysis of discrete-time dynamical systems
• Bifurcation analysis

Main Results:

• The stability domain of the null solution was determined.
• Existence of Fold/Cusp, Neimark-Sacker, and Flip bifurcations was proven.
• Resonant 1:3 and 1:4 bifurcations were identified.
• Marotto's chaotic behavior was theoretically proven under specific conditions.

Conclusions:

• The network's dynamics become increasingly complex as the characteristic parameter increases through bifurcation values.
• Sufficiently large interconnection coefficients and specific activation functions can induce chaotic dynamics.
• This research provides a theoretical framework for understanding complex behaviors in delayed neural networks.