# Special cubic perturbations of the Duffing oscillator $x"=x-x^3$ near the eight-loop

## TL;DR

This paper establishes an upper limit on the number of limit cycles emerging from the eight-loop of a Duffing oscillator under specific cubic perturbations, advancing understanding of nonlinear dynamic bifurcations.

## Contribution

It provides the first explicit upper bound for limit cycles near the eight-loop of the Duffing oscillator with special cubic perturbations.

## Key findings

- Upper bound for bifurcating limit cycles established
- Limit cycles analyzed near the eight-loop of the oscillator
- Perturbation effects on the oscillator's dynamics quantified

## Abstract

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+\lambda_{1}y+\lambda_{2}x^{2}+\lambda_{3}xy+\lambda_{4}x^{2}y . $$