# Zero-Hopf bifurcation in the general Van der Pol-Duffing equation

## TL;DR

This paper analytically investigates the coexistence of multiple periodic solutions and invariant tori in the general Van der Pol-Duffing oscillator, employing averaging methods and supporting findings with numerical examples.

## Contribution

It introduces an analytical approach using averaging methods to study complex dynamics in the Van der Pol-Duffing oscillator, including coexistence phenomena.

## Key findings

- Multiple periodic solutions coexist with invariant tori.
- Analytical results are validated through numerical examples.
- The study advances understanding of nonlinear oscillatory behavior.

## Abstract

We study analytically the coexistence of multiple periodic solutions and invariant tori in the general Van der Pol-Duffing oscillator equations. We use several results related to the averaging method in order to analytically obtain our results. We also provide numerical examples for all the analytical results that we provide.