Chapter 8.1, Problem 5E

Interpretation Introduction

Interpretation:

To determine whether the given statement is true or false. At any zero-eigenvalue bifurcation in two

dimensions, the nullclines always intersect tangentially.

Concept Introduction:

Nullclines are a set of points in the phase plane where $\dot{\text{x}}\text{ = }\dot{\text{y}}\text{ = 0}$.

The set of points in a phase plane where $\dot{\text{x}}\text{ = 0}$ is known as x-nullclines. These are the points where the vectors are directed either upward or downward.

The set of points in a phase plane where $\dot{\text{y}}\text{ = 0}$ is known as y-nullclines. These are the points where the vectors are directed horizontally.

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

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