Chapter 8.1, Problem 7E

Interpretation Introduction

Interpretation:

To find and classify all bifurcations for the system $\dot{\text{x}}\text{ = y - ax}$, $\dot{\text{y}}\text{ = - by + }\frac{\text{x}}{\text{1+x}}$.

Concept Introduction:

Fixed points are the intersection of nullclines $\dot{\text{x}}\text{ = 0}$, $\dot{\text{y}}\text{ = 0}$.

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

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