Phase-plane analysis
Objective. Classify equilibria and sketch trajectories.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Phase-plane analysis studies a two-dimensional autonomous system x′ = f(x,y), y′ = g(x,y) by plotting trajectories in the (x,y) plane rather than solving explicitly for time. Equilibria occur where f = g = 0, and their type is classified by the eigenvalues of the Jacobian matrix evaluated there: real eigenvalues of the same sign give a node, opposite signs give a saddle, and complex eigenvalues give a spiral or centre depending on the real part. For example, at an equilibrium with eigenvalues −1 and −3 the point is a stable node, so nearby trajectories converge into it along the eigenvector directions.
Worked example
In the phase plane for x′ = x, y′ = −y, classify the equilibrium at the origin.
- 1Find the Jacobian, which is diagonal with entries 1 and −1
- 2Eigenvalues are λ1 = 1 and λ2 = −1
- 3Opposite-sign real eigenvalues indicate a saddle point
- 4State the classification of the origin
Answer: Saddle point
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Practice
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Start practisingCommon mistake. Students often classify an equilibrium using only the sign of the trace without checking the determinant, which can wrongly label a saddle as a node. Always compute both eigenvalues of the Jacobian before deciding the type.
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