Chapter 7.5, Problem 2E

Interpretation Introduction

Interpretation:

To analyze the van der Pol oscillator for $\text{μ}>>1$ in the standard phase plane where $\dot{\text{x}}\text{ = y}$ and $\stackrel{.}{\text{y}}=-\text{X}-\text{μ}\left({\text{X}}^2-1\right)$ and the advantage of Lienard plane.

Concept Introduction:

From the expression of van der pol, the non-linear dynamic equation is ${\ddot{\text{x}}\text{ + μ}\dot{\text{x}}\text{(x}}^{\text{2}}\text{-1) + x = 0}$. Here, x is the position coordinate, and $\text{μ}$ is the scalar parameter.

The expression of $\ddot{\text{x}}$ and $\dot{\text{x}}$ is $\ddot{\text{x}}\text{ = }\frac{{\text{d}}^{\text{2}}\text{x}}{{\text{dt}}^{\text{2}}}$, $\dot{\text{x}}\text{ = }\frac{\text{dx}}{\text{dt}}$.

The Leonhard’s plane is the plane in which the non-linear dynamic equation is shown in the graph of $\text{xy}$ plane and in which vector field is vertical. This is called null cline.