Chapter 6.3, Problem 14E

Interpretation Introduction

Interpretation:

To classify the fixed point at the origin for the system ${\dot{\text{x}}\text{ = -y + ax}}^{\text{3}}$, ${\dot{\text{y}}\text{ = x + ay}}^{\text{3}}$ for all real values of the parameter $\text{a}$.

Concept Introduction:

Fixed points occur where $\dot{\text{x}}\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{A = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$

If $\text{Δ > 0}$, then the origin is always at the center, according to the linearization.

The Eigenvalue $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$