An Algorithm for Detecting Hopf-Birfucation Varieties of Nonlinear Polynomial Systems

TL;DR

This paper introduces an algorithm to identify Hopf-Bifurcation varieties in polynomial dynamical systems by constructing Lyapunov functions through sum-of-products polynomial expressions, enabling stability analysis and bifurcation detection.

Contribution

It presents a novel algorithm that uses polynomial sum-of-products expressions with parameters to detect Hopf-Bifurcation varieties in nonlinear polynomial systems.

Findings

Successfully identifies Hopf-Bifurcation points in polynomial systems.

Provides a method to construct Lyapunov functions ensuring stability.

Enables stability analysis through parameterized polynomial expressions.

Abstract

An algorithm is presented here, for discovering Hopf-Bifurcation varieties of polynomial dynamical systems. It is based on the expression of specific polynomials, as sums of products of first degree polynomials, with parametrical coefficients. By giving to these parameters certain values, we ensure the positiveness of some quantities, constructing thereby proper Lyapunov functions, which guarantee the stability of the equilibrium point. The points where the afore mentioned positiveness fails, define the Hopf-Bifurcation varieties upon discussion.