MAST20029 · Engineering Mathematics
Phase Portraits and Critical-Point Classification
Having solved ẋ = Ax, this chapter reads off the geometry. You classify the critical point from the eigenvalues — node, saddle, star, spiral or centre, stable or unstable — using the formula-sheet decision table, and you sketch and justify the phase portrait (straight-line orbits along eigenvectors, arrows from the sign of λ, axis-crossing slopes, behaviour as t → ±∞). Non-linear systems are handled by finding critical points and linearising with the Jacobian. This is the Q5-type sketch-and-justify question.
What this chapter covers
- 011. Classifying the origin from eigenvalues: node, saddle, star, spiral, centre
- 022. Stable vs unstable from the sign of the real parts of the eigenvalues
- 033. The formula-sheet decision table and geometric multiplicity g
- 044. Straight-line orbits along eigenvector directions, arrowed by sign of λ
- 055. Behaviour as t → +∞ and t → −∞ (which eigen-line orbits asymptote to)
- 066. Axis-crossing slopes from dy/dx = ẏ/ẋ (set ẋ=0 or ẏ=0)
- 077. Non-linear systems: find critical points by solving ẋ = ẏ = 0
- 088. Linearisation by the Jacobian at each critical point, then classify
Classify and sketch a saddle
- 2 marksEigenvalues of A = [[1, 3], [3, 1]]: λ = 1 ± 3, so λ₁ = 4 and λ₂ = −2. Opposite signs mean the origin is an unstable SADDLE.
- 2 marksEigenvectors: for λ = 4, w = (1, 1) (the line y = x); for λ = −2, w = (1, −1) (the line y = −x). These are the two straight-line orbits.
- 1 markDirections: along y = x orbits move AWAY from the origin (e^(4t) grows); along y = −x orbits move TOWARD the origin (e^(−2t) decays). As t → +∞ orbits asymptote to y = x; as t → −∞ they asymptote to y = −x.
- 1 markAxis-crossing slopes from dy/dx = ẏ/ẋ = (3x + y)/(x + 3y): on the y-axis (x = 0) slope = 1/3; on the x-axis (y = 0) slope = 3.
- 1 markSketch: two eigen-lines y = ±x with arrows, hyperbola-like trajectories curving between them, crossing the axes at the computed slopes.
Key terms
- Critical point
- A point where ẋ = ẏ = 0 (the system is at rest). For a linear system it is the origin; its type is read from the eigenvalues of A.
- Node / saddle / spiral / centre
- The critical-point types: node (real same-sign eigenvalues), saddle (real opposite signs), spiral (complex with nonzero real part), centre (pure imaginary), with star/improper sub-cases for repeated eigenvalues.
- Stability
- A critical point is asymptotically stable if all eigenvalue real parts are negative (orbits return), unstable if any is positive (orbits leave); a centre is (neutrally) stable with closed orbits.
- Straight-line orbit
- A trajectory lying along an eigenvector direction; its arrow points outward if the eigenvalue is positive and inward if negative. These lines organise the whole portrait.
- Phase portrait
- A sketch of representative trajectories in the (x, y) plane with arrows of motion, showing the eigen-lines, the spiral/rotational sense, and the long-time behaviour.
- Jacobian linearisation
- For a non-linear system, the matrix of partial derivatives J = [[∂ẋ/∂x, ∂ẋ/∂y], [∂ẏ/∂x, ∂ẏ/∂y]] evaluated at a critical point; its eigenvalues classify that point locally.
Phase Portraits and Critical-Point Classification FAQ
How do I classify a critical point from the eigenvalues?
Two real same-sign eigenvalues give a node (stable if both negative, unstable if both positive); real opposite signs give a saddle (always unstable); complex with nonzero real part give a spiral (stable if the real part is negative); pure imaginary give a centre. Repeated eigenvalues give star or improper nodes depending on the geometric multiplicity. The formula-sheet table is the decision rule.
How do I sketch the phase portrait?
Draw the eigen-lines (the eigenvector directions) as straight-line orbits and arrow them by the sign of their eigenvalue (outward for positive, inward for negative). Add representative curved trajectories that follow the long-time behaviour, mark the axis-crossing slopes from ẏ/ẋ, and indicate the rotational sense for spirals and centres.
What do I do with a non-linear system?
Find the critical points by solving ẋ = 0 and ẏ = 0 simultaneously. At each one, compute the Jacobian (the matrix of partial derivatives) and classify that point from the Jacobian's eigenvalues using the same table. Note that linearisation can be inconclusive for borderline cases such as a centre.
How do I get the slopes where orbits cross the axes?
Use dy/dx = ẏ/ẋ. On the y-axis set x = 0 in that ratio; on the x-axis set y = 0. These give the exact slopes at which trajectories cross the axes — compute them rather than eyeballing, since the marks are for justified detail.
Exam move
Lock in the eigenvalue-to-type table so classification is instant, then practise the sketch-and-justify ritual the exam rewards: name the type and stability, draw and arrow the eigen-lines, state the t → ±∞ behaviour, and compute axis-crossing slopes from ẏ/ẋ. Treat arrows as following the sign of λ on each eigen-line, and for spirals and centres determine the rotational sense from a test velocity vector. For non-linear systems, drill the find-critical-points-then-Jacobian routine and classify each point separately. Every sentence of justification is a mark, so write the reasoning, not just the picture.
