EXPLORATION OF NONLINEAR OSCILLATIONS IN POLYNOMIAL VECTOR FIELDS VIA HARMONIC BALANCE AND BIFURCATION DIAGRAMS

Abstract

We study polynomial vector fields exhibiting nonlinear oscillations, using harmonic balance to approximate periodic solutions and bifurcation diagrams to explore qualitative transitions. Illustrative examples demonstrate phenomena such as Hopf, period‑doubling, and saddle‑node bifurcations in systems of increasing complexity. Our results provide insight into the nature and stability of limit cycles in polynomial ODE systems.