DIFFERENTIAL EQUATIONS 06
Phase Lines and Stability
A phase line shows how solutions of an autonomous differential equation move between equilibrium states.
The main idea
For \(y'=f(y)\),
\[\boxed{f(y)\gt0\Rightarrow y\text{ increases}}\]\[\boxed{f(y)\lt0\Rightarrow y\text{ decreases}}\]
1. Equilibria
Equilibria satisfy \(f(y)=0\).
2. Example
For \(y'=y(2-y)\), equilibria are \(0\) and \(2\).
For \(y\lt0\), \(y(2-y)\lt0\). For \(0\lt y\lt2\), it is positive. For \(y\gt2\), it is negative.
3. Stability
Arrows point toward \(y=2\), so it is stable. They point away from \(y=0\), so it is unstable.
4. Semi-stability
An equilibrium may attract from one side and repel from the other.
5. Derivative test
\[\boxed{f'(y^*)\lt0\Rightarrow\text{locally asymptotically stable}}\]\[\boxed{f'(y^*)\gt0\Rightarrow\text{unstable}}\]
If \(f'(y^*)=0\), inspect the signs directly.
6. Long-term behaviour
A phase line often reveals equilibria, stability and long-term direction without requiring an explicit solution.
7. Practical method
\[\boxed{f(y)=0\rightarrow\text{divide into intervals}\rightarrow\text{check signs}\rightarrow\text{draw arrows}\rightarrow\text{classify equilibria}}\]