Chapter 7.5, Problem 6E

Interpretation Introduction

Interpretation:

In case of biased van der pol oscillator,

To find and classify all fixed points.

To plot nullclines in Leonard plane and to show that if nullclines intersect on the middle Branch of a cubic nullcline, then the fixed point is unstable.

For $\text{μ >> 1}$, system has stable limit cycle if and only if, $\left|\text{a}\right|{\text{< a}}_{\text{c}}$ is to be shown, where ${\text{a}}_{\text{c}}$ is to be determined.

The phase portrait for a slightly greater than a_c is to be sketched.

Concept Introduction:

Fixed point of a differential equation is a point where $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$

Nullclines are the curves where either $\dot{\text{x}}=0\text{ or }\dot{\text{y}}\text{ = 0}$. They show whether the flow is completely vertical or horizontal.

Vector fields in this aspect represent the direction of flow and whether flow is going away from the fixed point or coming towards it.

Phase portraits represent the trajectories of the system with respect to the parameters and give qualitative idea about evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

Van der pol oscillator is a simple harmonic oscillator with a nonlineardampingterm, which acts like ordinary damping term for $\left|\text{x}\right|\text{>1}$ and as a negative damping for $\left|\text{x}\right|\text{<1}$.