The Van der Pol Equation

TL;DR

This paper explores the theoretical and numerical analysis of the Van der Pol oscillator, including bifurcation diagrams, Fourier spectra, and chaotic attractors, and demonstrates how to generate sound files from these solutions.

Contribution

It combines theoretical methods with numerical simulations to analyze the Van der Pol oscillator's solutions, including chaos, and creates auditory representations of different dynamical states.

Findings

Identification of bifurcation structures and chaos in the Van der Pol oscillator.

Construction of Fourier spectra and sound files representing different solutions.

Evidence of period doubling cascades leading to chaos.

Abstract

In this work, we present the basic theoretical efforts that are known in order to deal with non-trivial solutions of the Van der Pol oscillator, such as theory of average, successive approximations and symbolic dynamics. We also construct a set of diagrams (bifurcation, 2D and 3D Fourier power spectra) and maps, based on numerical investigations, corresponding to the expected theoretical results. Furthermore we examine closely the existence of chaotic attractors, both theoretically (with symbolic dynamics) and numerically (period doubling cascades). We show how we constructed sound files, based on the Fourier spectra, each one corresponding to a periodic, an almost periodic and a chaotic solution, as one of the parameters of the system alters.