Chapter 5.1, Problem 11E

Interpretation Introduction

Interpretation:

By using definitions of different types of stabilities, the stabilities of fixed points are to be proved.

Concept Introduction:

A fixed point $x^{*}=0$ is said to be globally attracting, if $x^{*}$ attracts all trajectories in the phase plane.

Liapunov Stable: A fixed point is Liapunov stable, if all the trajectories start sufficiently close to $x^{*}$ and remains close to it for all positive time.

When a fixed point is Liapunov stable, but not attracting is called neutrally stable. The nearby trajectories are neither attracted nor repelled from neutrally stable points.