Chapter 7.6, Problem 21E

Interpretation Introduction

Interpretation:

To show that the frequency of the limit cycle for the van der pol oscillator ${\ddot{\text{x}}\text{ + ε(x}}^{\text{2}}\text{-1)}\dot{\text{x}}\text{ + x = 0}$ is $\text{ω = 1-}\frac{1}{16}{\text{ε}}^2+\text{O(}{\epsilon }^3)$ using Poincare-Lindstedt method.

Concept Introduction:

In perturbation theory, when regular perturbation approach fails, the technique of uniformly approximating periodic solution to ordinary differential equations is known as Poincaré-Lindstedtmethod. By this method, the secular terms are removed.

The equation of nonlinear system is:

$\ddot{\text{x}}\text{ + x + εh}\left(x,\dot{x}\right)\text{= 0}$

In Poincare-Lindstedt method, $\ddot{\text{x}}\text{ and }\dot{\text{x}}$ are expressed as follows:

$\ddot{\text{x}}\text{ = }\frac{{\text{d}}^{\text{2}}\text{x}\left(\text{τ}\left(\text{t}\right)\right)}{\text{dt}}{\text{= ω}}^{\text{2}}\frac{{\text{d}}^{\text{2}}\text{x}}{{\text{dτ}}^{\text{2}}}$

$\dot{\text{x}}\text{ = ω}\frac{\text{d x}\left(\text{τ}\left(\text{t}\right)\right)}{\text{dτ}}$

$\text{x = x}\left(\text{τ}\right)$

The equation for time period is $\text{τ = ωt}$.