Chapter 7.5, Problem 4E

Interpretation Introduction

Interpretation:

The system $\ddot{\text{x}}\text{+μf(x)}\dot{\text{x}}\text{+x = 0}$ is equivalent to $\dot{\text{x}}\text{ = μ}\left(\text{y-F}\left(\text{x}\right)\right)\text{, }\dot{\text{y}}\text{ = }\frac{\text{-x}}{\text{μ}}$ where $\text{F}\left(\text{x}\right)$ is piecewise linear function is to be shown, and the nullclines are to be graphed. The system exhibits relaxation oscillations for $\text{μ>>1}$ is to be shown as well. Also, the limit cycle in the $\text{(x,y)}$ plane is to be plotted, and the period of this limit cycle is to be estimated.

Concept Introduction:

The van der pol non $\text{-}$ linear dynamic equation is $\ddot{\text{x}}\text{ + μ}\dot{\text{x}}\left({\text{x}}^{\text{2}}\text{ - 1}\right)\text{ + x = 0}$

If oscillations contain repetition of extremely slow build up, followed by sudden discharge, then such oscillation are called relaxation oscillations.