To sketch a trajectory, plot the initial point and draw a smooth curve that is everywhere tangent to the direction indicated by the phase portrait. Add arrows showing how the solution moves as time increases.
This is AHL 5.17 Phase portraits, so it is HL-only content. For a linear coupled system
the phase portrait represents the solutions in the (xy)-plane rather than plotting either variable against time.
Method
- Identify the equilibrium point, where and . For the standard homogeneous linear system, this is usually .
| Eigenvalue pattern | Expected trajectory behaviour |
|---|---|
| Two negative real eigenvalues | Moves towards a stable node |
| Two positive real eigenvalues | Moves away from an unstable node |
| Real eigenvalues with opposite signs | Curves around a saddle point, approaching or leaving along eigenvector directions |
| Complex eigenvalues | Spirals; the sign of the real part determines inward or outward motion |
| Purely imaginary eigenvalues | Closed circular or elliptical trajectories |
A common misconception is that any curve through the initial point is acceptable. It is not: the curve must follow the local vector directions, and distinct trajectories of an autonomous system cannot cross because that would assign two directions to the same point.
Exam technique: Show the initial point, correct qualitative shape, direction arrows, and relevant asymptotic behaviour. If asked to justify the sketch, refer explicitly to the eigenvalues, eigenvectors, or signs of and ; a shape without reasoning may not earn full credit.