Introduction to Phase Portraits
A phase portrait is a visual tool used to understand the long-term behaviour of a system of coupled differential equations — without needing to solve the equations explicitly. Instead of plotting or against time, we plot the trajectory of the point directly in the -plane (called the phase plane).
In AHL 5.17, we focus on linear systems of the form:
\begin{cases} \dfrac{dx}{dt} = ax + by \6pt] \dfrac{dy}{dt} = cx + dy \end{cases}$$
where are real constants. This can be written compactly in matrix form:
The matrix is called the coefficient matrix of the system, and its eigenvalues and eigenvectors completely determine the qualitative behaviour of the phase portrait.
Think of the phase portrait like a wind map. At every point in the plane, the system tells you the "wind direction" — the direction and speed at which the state of the system is moving. The trajectories are the paths a leaf would follow if dropped into that wind field.
Equilibrium Points
A point where and simultaneously. At an equilibrium point, the system is stationary — there is no change over time.
For a linear system , if is invertible (i.e. ), the only equilibrium point is the origin .
The nature of this equilibrium — whether trajectories approach it, move away from it, or orbit around it — depends entirely on the eigenvalues of .
For the system:
\begin{cases} \dfrac{dx}{dt} = 2x - y \6pt] \dfrac{dy}{dt} = x + y \end{cases}$$
Set both equations to zero: and . Solving gives . The only equilibrium point is .
In AHL 5.17, we only consider systems with distinct, non-zero eigenvalues. This ensures the origin is always the unique equilibrium point and covers the most instructive range of behaviours.