Chapter 2.4, Problem 5E

Interpretation Introduction

Interpretation:

Linear stability analysis is to be used to classify the fixed points of the given system. If linear stability analysis fails, the graphical argument is to be used to decide the stability.

Concept Introduction:

Fixed points are the points where $\dot{\text{x}}\text{ = 0}$.

The fixed point, ${\text{x}}^{\text{*}}$ of the system, is stable if ${{\text{f}}^{\prime }\text{(x}}^{\text{*}}\text{) < 0}$, is unstable if ${{\text{f}}^{\prime }\text{(x}}^{\text{*}}\text{) > 0}$, and the stability of the point is inconclusive if ${{\text{f}}^{\prime }\text{(x}}^{\text{*}}\text{) = 0}$.

The flow is towards the right when $\dot{\text{x}}\text{ > 0}$ and, it is towards the left when $\dot{\text{x}}\text{ < 0}$.

The point where the flow is toward it is called the stable point, and the point where the flow is away from it is called the unstable point.