Graph Theoretic Framework of Dynamical Systems Through the Boundary Polynomial

TL;DR

This paper explores the connection between graph theory and the qualitative analysis of 2D dynamical systems using boundary polynomials, highlighting implications for understanding limit cycles and Hilbert's 16th problem.

Contribution

It introduces a graph theoretic framework based on boundary polynomials to analyze dynamical systems and their limit cycles.

Findings

Reinterprets classical results via boundary polynomials

Provides new insights into limit cycle analysis

Links graph theory with Hilbert's 16th problem

Abstract

This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results through the lens of boundary polynomials of graphs. The theory naturally leads to questions about limit cycles, which arise in many processes in nature and remain a central object of study in dynamical systems. The significance of limit cycles is underscored by their place in Hilbert's 16th problem, one of the unsolved challenges from his famous list of 23 problems posed in 1900.