Chapter 5.1, Problem 10E

Interpretation Introduction

Interpretation:

To determine whether the origins of the given systems of differential equations are attracting, Liapunov stable, asymptotically stable or none of these.

Concept Introduction:

Point $\text{x*}$ of a system $\dot{\text{x}}\text{ = f(x)}$ is attracting, if any trajectory that starts within a distance $\text{δ}$ of $\text{x*}$ isguaranteed to converge to $\text{x*}$ eventually.

I.e. $\text{x*}$ is attracting, if there is a $\text{δ> 0}$, such that $\underset{\text{t}\to \infty }{\lim }\text{x(t) = x*}$, whenever $\Vert \text{x(0) - x*}\Vert \text{ < δ}$.

Point is Liapunov stable, if nearby trajectories remain close for alltime.

I.e. for each $\text{ε> 0}$, there is a $\text{δ> 0}$, such that $\Vert \text{x(0) - x*}\Vert \text{ < ε}$, whenever $\Vert \text{x(0) - x*}\Vert \text{ < δ}$.

$\text{x*}$ isasymptotically stable, if it is both attracting and Liapunov stable.