Chapter 7.6, Problem 16E

Interpretation Introduction

Interpretation:

Consider the van der Pol oscillator $\ddot{\text{x}}\text{ + }\epsilon {\dot{\text{x}}\text{(x}}^2\text{-1) + x = 0 }$ in the limit $\epsilon <<1$. Assume the limit cycle is a circle of unknown radius $\text{a}$ about the origin. Using the normal form of Green’s theorem ($\text{2-D}$ divergence theorem) ${\displaystyle \underset{C}{\oint }\text{v}\text{.n dl}}={\displaystyle {\iint }_A\nabla .\text{v }}\text{dA}$, evaluate the integral, and show that $\text{a}\approx 2$ by substituting $\text{v = }\dot{\text{x}}\text{= (}\dot{x},\dot{y})$ where $C$ is the cycle and $A$ is the region enclosed.

Concept Introduction:

• ➢ Use Dulc’s criteria for the third method for ruling out closed orbit based on Green’s theorem.

• ➢ Select the real value function $\text{g(x)}$ and obtain $\nabla .(\text{g}\dot{\text{x}}\text{)}$

• ➢ Apply Green’s theorem.

• ➢ Determine circle radius.