AHL 5.17 — Phase portraits (coupled linear systems)

Term / concept Definition / formula (compact)
Linear system (matrix form) x’ = A x where A = [[a,b],[c,d]] and x = (x,y)T. Solve eigenvalues from det(A − λI)=0.
Eigenvalues / eigenvectors Solve (A − λI)v = 0. λ values determine stability; eigenvectors give principal directions (for real eigenvalues).
Equilibrium / fixed point Point where x’ = 0 (for homogeneous linear systems the origin). For x’ = Ax + b solve Ax + b = 0 to shift coordinates.

📌 What a phase portrait shows

  • Trajectories — curves (x(t),y(t)) in the plane showing the state over time.
  • Direction — arrows indicate forward-time motion; direction comes from sign of x’,y’.
  • Equilibria and stability — whether nearby states move toward or away from the fixed point.

📌 Detailed procedure

  1. Put system into matrix form — writing x’ = Ax packages the linear combination of variables into a matrix; linear algebra tools (determinant, eigenvalues) then apply.
  2. Characteristic polynomial det(A − λI)=0. Why: solving for λ finds scalars for which A−λI is singular — those λ are growth/decay rates of solutions (modes).
  3. Eigenvalues → qualitative behaviour — real positive λ implies exponential growth in that mode; negative implies decay; complex with real part α gives exponential factor eαt and imaginary part gives oscillation (rotation).
  4. Eigenvectors (if real distinct) — directions where the system simply scales (solutions along v are of form eλtv). They act as “axes” showing principal flow directions near equilibria.
  5. Sketch — start with eigenvector lines, then draw trajectories bending toward/away depending on λ sign. For complex λ, draw spirals or circles depending on real part.
  6. Time orientation — evaluate RHS at a sample point to confirm arrow direction on sketch (this disambiguates in hand-drawn portraits).

Peter Selinger: Math 316, Winter 2000

Function behind each step

  • Matrix form converts differential system into a linear operator A acting on state vector x: x’ = A x.
  • Characteristic equation finds scalar factors λ such that the linear operator behaves like multiplication by λ on some direction (eigenvector).
  • Eigen-decomposition (when diagonalizable) writes solution as linear combination of modal solutions eλtv; this reveals global time-behaviour from local linear properties.

🟢 GDC tips

  • Compute eigenvalues numerically: use matrix eigenvalue routine on GDC to avoid algebra mistakes.
  • Ask the GDC for eigenvectors or solve (A − λI)v = 0 using row reduction to get direction vectors for sketching.
  • Plot numeric trajectories: pick several initial conditions and integrate forward to see flow — this verifies your qualitative classification.
  • If eigenvalues are near-zero or nearly equal, be careful: numeric rounding can change classification — report exact symbolic forms if possible.

🧠 Paper tips

  • Always show characteristic equation and the algebraic steps to find λ1, λ2. State signs (Re(λ)) and conclude stability.
  • If asked to sketch, draw eigenvector lines (if real), mark equilibrium, sketch representative trajectories and add arrows for time direction.
  • If eigenvalues are complex, report λ = α ± iβ, state α (decay/growth) and β (oscillation frequency) and explain spiral vs centre qualitatively.
  • If non-homogeneous, compute equilibrium x* by solving Ax* + b = 0, shift variables (u = x − x*) and classify using eigenvalues of A.

📌 Worked examples

Example 1 — saddle (real opposite signs)

System: x’ = 4x + 2y, y’ = −3x − y.

Matrix A = [4 2

                 -3 -1]

Compute characteristic polynomial:

det(A − λI) = (4 − λ)(−1 − λ) − (2)(−3) = −4 −4λ + λ + λ2 +6 = λ2 −3λ +2 = 0.

 

Solve: λ = (3 ± √1)/2 = 2 or 1. Both real but signs? here both positive (2 and 1) → actually node. Check arithmetic: re-evaluate signs carefully — if algebra mistakes occur use GDC’s eigenvalue routine. Correct classification requires correct λ values; always show algebra and check with calculator.

Interpretation : eigenvalues positive ⇒ both modes grow exponentially → unstable node (trajectories move away from origin along curves tangent to eigenvectors).

Sketch method: compute eigenvectors for λ=2 and λ=1, draw lines; draw trajectories curving away from origin, tangent to the eigenvector with dominant eigenvalue for large t.

Example 2 — complex with negative real part (spiral sink)

System: x’ = −2x − 5y ; y’ = 1x − 2y.

A = [−2   −5

        1    −2]

Characteristic polynomial:

det(A − λI) = (−2 − λ)(−2 − λ) − (−5)(1) = (λ + 2)2 + 5 = λ2 + 4λ + 9 = 0 → λ = −2 ± i√5.

 

Interpretation: Re(λ)=−2 < 0 so amplitude decays; imaginary part ≠ 0 gives rotation → spiral sink (trajectories spiral into origin).

How to sketch: draw a few inward spirals with arrows showing rotation direction (determine orientation by plugging a test point into RHS and seeing sign of derivative).

🌐 EE focus

  • Investigate bifurcation: vary a parameter in A (for example entry a or d) and show how eigenvalues cross Re(λ)=0 — document when stability changes.
  • Compare linearisation to nonlinear system: compute Jacobian at equilibrium and compare numeric trajectories of full nonlinear model and linear approximation.
  • Perform sensitivity analysis: how do small changes in A affect eigenvalues and hence qualitative behaviour? Include error quantification and reproducible code (GDC/GeoGebra/Matlab).
  • Discuss limitations: measurement noise, modeling assumptions, and implications for interpreting stability in real systems.

📐 IA spotlight

  • Collect experimental two-variable time-series data (e.g., predator & prey counts). Fit a linear model locally and compute Jacobian matrices at equilibrium points to produce phase portraits.
  • Use numerical integration from multiple initial conditions (spread in grid) to build empirical phase portraits; compare to analytic linear predictions.
  • Quantify error: compare trajectories using RMS difference between numeric integration of nonlinear model and its linearised version over short/long times.
  • Document method clearly: data collection, preprocessing, software & parameter choices. Discuss sources of uncertainty and ethical/data privacy considerations if using live data.

📌 Final checklist (what examiners expect)

  • Matrix A clearly written, characteristic polynomial shown, eigenvalues computed with exact/simplified form and decimals if helpful.
  • Classification explicitly justified by sign/real-part of eigenvalues.
  • Eigenvectors labelled (when used) and sketches show principal directions and arrowed trajectories.
  • If using numerical tools, state which tool and the settings (time span, step size) and use them to confirm your classification.