Chapter 7.3, Problem 7E

Interpretation Introduction

Interpretation:

To rewrite the given system in polar coordinates. There is at least only limit cycle, and that if there are so many, they all have the similar period $\text{T}\left(\text{a,b}\right)$ is to be proved. Also prove that for $\text{b=0}$, there is only one limit cycle.

Concept Introduction:

Poincare $\text{-}$ Bendixson theorem states that if a dynamical system in which $\dot{\text{x}}\text{ = f (x, y), }\dot{\text{y}}\text{ = g (x, y)}$ is continuously differentiable, and if it remains in a closed, bounded region $\text{R}$ formed by trajectory of a vector field on the plane for all $\text{t}\le \text{0}$, then the trajectory must form a periodic orbit.

Poincare $\text{-}$ Bendixson Theorem states that if there are no fixed points in a closed, bounded region formed by the trajectory of a vector field on the plane, then trajectory must eventually approach a closed orbit.

If there are no fixed points in the trapping region R, then according to the Poincare $\text{-}$ Bendixson Theorem, R contains a closed orbit.