Chapter 7.3, Problem 10E

Interpretation Introduction

Interpretation:

For two dimensional system ${\dot{\text{x}}\text{ = Ax - r}}^{\text{2}}\text{x}$, where $\text{r = ||x|| and A}$ is $\text{2 × 2}$ constant real matrix with complex eigen values $\text{α ± iω}$, the existence of at least one limit cycle for $\text{α > 0}$ and its nonexistence for $\text{α < 0}$ is to be shown.

Concept Introduction:

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

The method for ruling out closed orbits, that is, ruling out the existence of a limit cycle based on Green’s theorem is called Dulac’s criterion.

Consider a continuously differentiable vector field $\dot{\text{x}}\text{ = f(x)}$ explained on a subset $\text{R}$ of the plane. If a continuously differentiable function, real valued function $\text{g(x)}$ and if $\nabla \text{.(g(}\dot{\text{x}}\text{))}$ has same sign throughout subset R, then there will not be any closed orbit lying entirely in $\text{R}$.