Qualitative dynamics

Objective. Use equilibria and phase lines to analyse behaviour.

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Learn: the key idea

Qualitative dynamics studies autonomous equations dy/dt = f(y) without solving them explicitly, using equilibria (where f(y)=0) and the sign of f(y) between them to sketch a phase line. Mark equilibria on a y-axis, then test f(y) in each interval: positive means y increases (arrow up), negative means y decreases (arrow down). An equilibrium is stable if arrows point towards it from both sides and unstable if they point away. For dy/dt = y(1−y), equilibria are y=0 and y=1; testing shows y=0 is unstable and y=1 is stable, so solutions starting near 1 converge to it.

Worked example

For the autonomous equation dy/dt = y(2−y), classify the equilibrium at y = 2 using phase-line analysis.

1 $Equilibria where f(y)=0: y=0 and y=2$
2 $For y slightly below 2, f(y)=y(2-y)>0, so y increases$
3 $For y slightly above 2, f(y)<0, so y decreases$
4 $Arrows point towards y=2 from both sides, so it is stable$

Answer: y = 2 is a stable equilibrium

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Common mistake. Assuming every equilibrium is stable just because it is a solution; stability depends on the sign of f(y) nearby, not on the equilibrium existing. Always test both sides before labelling it stable.

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