Phase Portraits & Coupled DEs (AI HL)
A coupled system of two linear differential equations describes how two quantities change together over time - a predator and its prey, two connected populations, two currents in a circuit. Rather than solving for exact formulas, you read the long-term behaviour straight off the eigenvalues of the coefficient matrix and sketch it as a phase portrait. This page covers the stability rules, how to sketch trajectories, and the mistakes that lose marks. It's part of the broader Differential Equations topic.
13 questions on this sub-topic.
Reading stability from eigenvalues
Covered under IB syllabus reference AHL5.17: phase portraits for coupled systems of the form \(\dfrac{dx}{dt}=ax+by,\ \dfrac{dy}{dt}=cx+dy\), with distinct, non-zero eigenvalues. These aren't formula-booklet entries - they're rules you apply after finding the eigenvalues of the coefficient matrix.
The system
\(\dfrac{dx}{dt}=ax+by,\quad\dfrac{dy}{dt}=cx+dy\)
Write the coefficients as a matrix and find its eigenvalues - they alone determine the shape and stability of the origin's equilibrium.
Stability rules
Real, same sign → node. Real, opposite sign → saddle. Complex → spiral. Purely imaginary → centre (circle/ellipse).
Stable when every eigenvalue (or its real part, for complex pairs) is negative; unstable if any is positive.
Need the full syllabus wording and the wider calculus formula table? See Differential Equations.
Worked examples
A system \(\dot{\mathbf{x}} = A\mathbf{x}\) has eigenvalues \(\lambda_1 = 2 + 3i\) and \(\lambda_2 = 2 - 3i\).
(a) State the type of equilibrium at the origin.
(b) State whether it is stable or unstable.
(c) Give a reason for your answer to (b).
Worked solution
(a) Complex eigenvalues with non-zero imaginary part. M1
\(\Rightarrow\) the equilibrium is a spiral (focus). A1
(b) Unstable. A1
(c) Real part \(= 2 > 0\), so trajectories spiral outward. M1
A system of coupled differential equations has the form
\[\frac{dx}{dt} = ax + by, \quad \frac{dy}{dt} = cx + dy.\]
The eigenvalues of the coefficient matrix are \(\lambda_1 = 3\) and \(\lambda_2 = -1\).
(a) State the type of equilibrium point at the origin.
(b) Sketch a phase portrait showing the typical behaviour of solution trajectories.
(c) State whether the origin is stable, unstable or semi-stable.
Worked solution
(a) Saddle point (one positive, one negative real eigenvalue). A1
(b) Trajectories approach the origin along one axis (direction of \(\lambda_2 = -1\)) and move away along the other (direction of \(\lambda_1 = 3\)), forming a hyperbolic pattern. Award marks for correct saddle-point shape A1
with correct arrows. A1
(c) Unstable (solutions move away from origin along the \(\lambda_1\) direction). A1
Common mistakes
- Judging stability from the eigenvalues themselves, not their sign. A larger-magnitude eigenvalue doesn't mean "more unstable" - only the sign (or, for complex pairs, the sign of the real part) decides stable versus unstable.
- Forgetting the centre case. Purely imaginary eigenvalues give closed loops around the origin - trajectories neither approach nor escape, so the equilibrium is stable but not asymptotically stable. Don't lump this in with the spiral case.
- Drawing saddle-point arrows the wrong way round. Trajectories move toward the origin along the eigenvector for the negative eigenvalue and away along the eigenvector for the positive one - mixing these up gives an unstable-looking sketch for what should read as mixed.
Ready to practise properly?
13 phase-portrait questions, marked instantly like the real exam.
Quick answers
How do eigenvalues determine the type of equilibrium?
Real eigenvalues of the same sign give a node (stable if both negative, unstable if both positive); real eigenvalues of opposite sign give a saddle point; complex eigenvalues give a spiral (stable if the real part is negative); purely imaginary eigenvalues give a closed loop (centre).
What is the general form of the coupled system?
\(\dfrac{dx}{dt} = ax + by\) and \(\dfrac{dy}{dt} = cx + dy\), where the coefficient matrix's eigenvalues determine the equilibrium type and stability at the origin.
Where do I do this on my GDC?
See the Using your GDC section on the Differential Equations page for calculator-specific steps.