# Morse Index Classification and Landscape of Kuramoto System for Hebbian-based Binary Pattern Recognition

## TL;DR

This paper analyzes the stability landscape of the Kuramoto model with Hebbian learning for binary pattern recognition, focusing on Morse index classification to understand how stable and unstable critical points affect system dynamics.

## Contribution

It introduces a systematic Morse index classification of critical points in the Kuramoto model, revealing the role of saddle points in the energy landscape for pattern recognition.

## Key findings

- Stable critical points store memorized patterns.
- Index-1 saddle points act as transition states.
- Multiple stable and unstable states influence convergence.

## Abstract

This study examines the Kuramoto model with a Hebbian learning rule and second-order Fourier coupling for binary pattern recognition. The system stores memorized binary patterns as stable critical points, enabling it to identify the closest match to a defective input. However, the system exhibits multiple stable states and thus the dynamics are influenced by saddle points and other unstable critical points, which may disrupt convergence and recognition accuracy. We systematically classify the stability of these critical points by analyzing the Morse index, which quantifies the stability of critical points by the number of unstable directions. The index-1 saddle point is highlighted as the transition state on the energy landscape of the Kuramoto model. These findings provide deeper insights into the stability landscape of the Kuramoto model than the stable equilibria, enhancing its theoretical foundation for binary pattern recognition.