Chapter 7.5, Problem 1E

Interpretation Introduction

Interpretation:

To show that the positive branch of the cubic nullcline begins at ${\text{x}}_{\text{A}}\text{ = 1}$ and ends at ${\text{x}}_{\text{B}}\text{ = 1}$ for the van der Pol oscillator with $\text{μ >>1}$.

Concept Introduction:

Nullclines are a set of points in the phase plane where $\dot{\text{x}}\text{ = }\dot{\text{y}}\text{ = 0}$.

The set of points in a phase plane where $\dot{\text{x}}\text{ = 0}$ is known as x-nullclines. The geometrical meaning of x-nullclines is the points where the vectors are pointed either straight upward or downward.

The set of points in a phase plane where $\dot{\text{y}}\text{ = 0}$ is known as y-nullclines. The geometrical meaning of y-nullclines is the points where the vectors are pointed horizontally.

To find the equation for x-nullclines and y-nullclines, put $\dot{\text{x}}\text{ = }\dot{\text{y}}\text{ = 0}$ and solve for

$\text{x = f(x,y) and y = f(x,y)}$.