Differential Equations › Section E · Systems of differential equations › Topic E.4
The phase plane
A solution of a two-unknown system is a pair of functions, but it can also be drawn as one curve traced out in the plane. Collect enough of those curves and a single picture shows whether every solution settles down, runs away or circles forever, and the eigenvalues of the matrix predict that picture.
In this topic
- Solutions as moving points
- The equilibrium at the origin
- Sketching from tangent vectors
- Worked example: reading a saddle from arrows
- Words for stability
- Real eigenvalues: nodes and saddles
- Complex eigenvalues: spirals and centres
- The trace-determinant test
- Worked example: three quick classifications
- Stability in any dimension
- Common mistakes to avoid
- Key takeaways
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