# Van der Pol - Duffing oscillator: global view of metamorphoses of the amplitude profiles

## TL;DR

This paper analyzes the global bifurcation structure of the amplitude profiles of the Duffing--Van der Pol oscillator, identifying singular points where dynamic metamorphoses occur, using the Krylov-Bogoliubov-Mitropolsky approach.

## Contribution

It provides a comprehensive computation of the bifurcation set and illustrates the metamorphoses of dynamics at singular points of the amplitude profile.

## Key findings

- Identification of singular points on the amplitude curve.
- Computation of the bifurcation set in parameter space.
- Examples of dynamic metamorphoses at singular points.

## Abstract

Dynamics of the Duffing--Van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude $A$ on forcing frequency $\Omega $ as an implicit function, $F\left( A,\Omega \right) =0$, referred to as resonance curve or amplitude profile. In singular points of the amplitude curve the conditions $\frac{\partial F}{\partial A}=0$, $\frac{\partial F}{\partial \Omega }=0$ are fulfilled, i.e. in such points neither of the functions $A=f\left( \Omega \right) $, $\Omega =g\left( A\right) $, continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed. Several examples of singular points and the corresponding metamorphoses of dynamics are presented.