Chapter 7.2, Problem 14E

Interpretation Introduction

Interpretation:

The system ${\dot{\text{x}}\text{ = x}}^2\text{ - y - 1}$, $\dot{\text{y}}\text{ = y(x - 2)}$ has three fixed points is to be shown and these points should be classified. The system has no closed orbits is to be shown by considering the three straight lines through pairs of fixed points as well. Also, phase portrait for the system is to be sketched.

Concept Introduction:

Fixed points of a system are the points where $\dot{\text{x}}\text{ = 0}$, $\dot{\text{y}}\text{ = 0}\text{.}$

If $\det ({\rm A})<0$ andeigenvalues are real and have opposite signs, then the fixed points of the system are called saddle points.

If $\det ({\rm A})>0$ and eigenvalues are both positive signs, then the fixed points of the system are called unstable fixed points.

If $\det ({\rm A})>0$ is and eigenvalues are both negative signs, then the fixed points of the system are called stable fixed points.

The plotted trajectories of the dynamic system are called phase portrait.

A phase portrait gives the qualitative behavior of the solutions of a differential equation.