Quasi-periodic solutions for differential equations with an elliptic-type degenerate equilibrium point under small perturbations

TL;DR

This paper investigates the existence of quasi-periodic solutions in differential equations with elliptic-type degenerate equilibrium points under small quasi-periodic perturbations, extending results to both ODEs and DDEs, including the delayed van der Pol oscillator.

Contribution

It establishes the existence of quasi-periodic solutions near degenerate equilibria for perturbed ODEs and DDEs, including applications to specific oscillators.

Findings

Existence of quasi-periodic solutions near equilibrium points.

Application to delayed van der Pol oscillator with zero-Hopf singularity.

Results hold for most parameter values under certain hypotheses.

Abstract

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity.