# Computing time average bounds for the Van der Pol oscillator in Julia

22 February 2021 | R. Rosa

## Introduction

We addressed, in the previous Time average bounds via Sum of Squares post, the problem of estimating the asymptotic limit of time averages of quantities related to the solutions of a differential equation.

Here, the aim is to consider an example, namely the Van der Pol oscillator, and use two numerical methods to obtain those bounds: via time evolution of the system and via a convex semidefinite programming using Sum of Squares (SoS), both discussed in the previous post.

This example is addressed in details in Fantuzzi, Goluskin, Huang, and Chernyshenko (2016). My main motivation is to visualize the auxiliary function that yields the optimal bound via SoS. That is the main reason to choose a two-dimensional system.

That bound depends on the chosen degree $$m$$ for the auxiliary function appearing in the SoS method. Here are the results for specific values of $$m$$: