Section 19.8

Equilibrium Solutions

For Autonomous equations , we can predict the entire qualitative behavior without solving a single integral.

Introduction

An Autonomous DE has no explicit time dependence: .

The constant solutions occur where . These are called Equilibrium Solutions or Critical Points.

Classification of Stability

Stable (Sink)

Solutions approach from both sides.
.

Unstable (Source)

Solutions move away from on both sides.
.

Semi-Stable (Node)

One side approaches, one side leaves.
?

Visual: Phase Line Portrait

Interactive: Logistic Growth Phase Line

Worked Examples

Example 1: Finding Equilibria

Find and classify equilibria for .

1. Find roots:

Equilibria: .

2. Check Signs:

: (+)(+) > 0 (Increasing).
: (-)(+) < 0 (Decreasing).
: (-)(-) > 0 (Increasing).

3. Classify:

: Source (Unstable) because arrows go away.

: Sink (Stable) because arrows go towards.

Example 2: Semi-Stable

Classify equilibria for .

1. Roots:

and .

2. Signs:

: (+)(+) > 0.
: (+)(+) > 0.
: (+)(-) < 0.

3. Classify:

: Up then Up. Semi-Stable.

: Down then Up. Unstable (Source). Wait: Below -1 it decreases (away). Above -1 it increases (away). Yes, Unstable.

Example 3: Parameter Dependence

Analyze equilibria for as changes.

1. Find Roots:

2. Cases:

: No real roots. No equilibria. is always positive.
: . One root at (Semi-Stable).
: Two roots and .

3. Bifurcation:

As crosses 0, two equilibria appear "out of nowhere". This is called a Bifurcation.