Section 22.6

Phase Plane

We don't always need and . Sometimes we just want to know: where does the flow go?

Introduction

In the Phase Plane, we plot vs (or vs ).
The variable becomes the parameter along the curve.

Real eigenvalues of same sign.
Positive = Source (Unstable).
Negative = Sink (Stable).

Real eigenvalues of opposite signs.
Always Unstable.

Complex eigenvalues.
Real part determines stability.

Pure imaginary eigenvalues.
Orbits are closed loops.

The geometry of the solution depends entirely on Trace () and Determinant ().

Classify .

1. Find Eigenvalues:

2. Analyze:

Both negative real. This is a Stable Node (Sink).

Classify .

1. Eigenvalues:

2. Analyze:

Opposite signs. This is a Saddle Point (Unstable).

Flow approaches origin along y-axis, leaves along x-axis.

Classify .

1. Eigenvalues:

2. Analyze:

Pure imaginary. This is a Center (Neutrally Stable).

Trajectories are ellipses.