A phase portrait is a diagram showing how the solutions of a system of coupled differential equations evolve in the phase plane. Each curve represents a possible solution, while arrows show its direction as time increases.
For the AHL 5.17 system
A phase portrait is a diagram showing how the solutions of a system of coupled differential equations evolve in the phase plane. Each curve represents a possible solution, while arrows show its direction as time increases.
For the AHL 5.17 system
the horizontal axis represents and the vertical axis represents . Therefore, each point ((x,y)) describes the complete state of the system at one moment, rather than showing either variable directly against time.
At every point, the vector
gives the direction in which the state moves. Following these directions produces trajectories. The origin is an equilibrium point because both derivatives equal zero there.
The matrix of the system is
Its eigenvalues determine the main shape and stability of the phase portrait.
| Eigenvalue behaviour | Phase portrait near the origin |
|---|---|
| Two negative real eigenvalues | Stable node: trajectories move towards the origin |
| Two positive real eigenvalues | Unstable node: trajectories move away from the origin |
| Real eigenvalues with opposite signs | Saddle point: some trajectories approach and others move away |
| Complex eigenvalues with negative real part | Stable spiral towards the origin |
| Complex eigenvalues with positive real part | Unstable spiral away from the origin |
| Purely imaginary eigenvalues | Closed circles or ellipses around the origin |
For example, and have eigenvalues and . Since they have opposite signs, the origin is a saddle point.
A common misconception is that a phase portrait is a graph against time. It is not: it plots against , with time represented only by arrow direction.
Exam technique: In an IB Math AI HL response, identify the eigenvalue type, classify the equilibrium, state whether trajectories move towards or away from the origin, and include the correct directional arrows. This is AHL 5.17, an HL-only topic.