Chapter 8.1, Problem 13E

Interpretation Introduction

Interpretation:

To find and classify all the fixed points. Nondimensionalize the given system. Also sketch all the qualitatively different phase portraits that occur as the dimensionless parameters are varied. Plot the stability diagram for the system. What types of bifurcations occur?

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. The geometrical meaning of x-nullclines is the points where the vectors are pointed either straight upward or downward.

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

The saddle-node bifurcation is the basic mechanism for the creation and destruction of fixed points.