Chapter 2.4, Problem 7E

Interpretation Introduction

Interpretation:

The analysis of the nonlinear system using the linear stability to classify the fixed points of the system ${\dot{\text{x}}\text{ = ax - x}}^{\text{3}}$, where $\text{ a}$ can be positive, negative or zero. If linear stability analysis fails, the graphical arguments are used to determine the stability.

Concept Introduction:

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

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