Sketching a Phase Portrait by Hand

From Four Skills to One Picture

Across this rung you have collected four separate skills. Guide 1 taught you to read a planar system x' = f(x, y), y' = g(x, y) as a velocity vector field, with trajectories threading along the arrows. Guide 2 located the equilibria and showed that near each one you can linearize using the Jacobian matrix. Guide 3 named the four local pictures — node, saddle, spiral, center — and guide 4 packed them into the trace-determinant plane, so two numbers classify the linear behaviour. This final guide does no new theory. It welds those four skills into one hand routine.

Why bother sketching by hand when a computer plots a flawless field in a second? Because most nonlinear systems have no closed-form solution at all — the qualitative picture is often the *only* answer you can get. A good hand sketch shows you what to expect, catches the equilibria a numerical plot might step over, and builds the intuition that lets you read a computer's output critically rather than trust it blindly. The goal is not a pretty drawing; it is understanding the long-term fate of every trajectory at a glance.

The Six-Step Routine

Here is the whole method as one ordered checklist. Work it top to bottom on every problem; each step hands the next one exactly what it needs. We will then walk a concrete example through all six.

1. Find the equilibria. Set x' = 0 and y' = 0 simultaneously and solve. These fixed points are the skeleton of the whole portrait.
2. Draw the nullclines. The x-nullcline is the curve f(x, y) = 0 (arrows are vertical there); the y-nullcline is g(x, y) = 0 (arrows are horizontal there). They cross exactly at the equilibria and carve the plane into regions of definite arrow direction.
3. Mark the flow direction in each region. Pick one easy test point per region, evaluate the signs of x' and y', and draw a little arrow (right/left, up/down). This is the coarse global flow.
4. Classify each equilibrium. Compute the Jacobian matrix there, then read its trace and determinant. Locate the point in the trace-determinant plane to name it: node, saddle, spiral, or center.
5. Add local detail at each equilibrium. For a node or saddle, draw the eigenvector directions (the straight-line solutions) and the inflow/outflow arrows along them. For a spiral, fix the sense of rotation; for a center, draw a small closed loop.
6. Connect the local pictures into global trajectories. Draw smooth curves that obey the region arrows, leave or approach each equilibrium correctly, and never cross one another. The result is the phase portrait.

A Worked Example: A Competing Pair

Take the system x' = x(3 - x - 2y), y' = y(2 - x - y), a simple competition model with x, y >= 0. Run the routine. Step 1, the equilibria: x' = 0 gives x = 0 or 3 - x - 2y = 0, and y' = 0 gives y = 0 or 2 - x - y = 0. Pairing these up yields four fixed points: (0, 0), (3, 0), (0, 2), and the interior crossing (1, 1). Step 2, the nullclines are the lines x = 0, 3 - x - 2y = 0 (for x') and y = 0, 2 - x - y = 0 (for y') — sketch those four lines and the four equilibria sit at their intersections.

Step 3, test the regions. At a tiny point like (0.1, 0.1) both x' and y' are positive, so the flow there points up-and-to-the-right — away from the origin. Step 4, classify with the Jacobian J = [3 - 2x - 2y, -2x; -y, 2 - x - 2y]. At (0, 0) it is [3, 0; 0, 2]: trace 5, determinant 6, both eigenvalues positive — an unstable [[node|node]] (a source). At (3, 0): [-3, -6; 0, -1], a stable node. At (0, 2): [-1, 0; -2, -2], also a stable node. At the interior point (1, 1): J = [-1, -2; -1, -1], trace -2, determinant 1 - 2 = -1 < 0 — a negative determinant, which is the unmistakable signature of a [[ode-saddle-point|saddle point]].

equilibrium     trace   det     type
  (0, 0)            5       6     unstable node (source)
  (3, 0)           -4       3     stable node
  (0, 2)           -3       2     stable node
  (1, 1)           -2      -1     saddle  (det < 0)

Steps 5 and 6 finish the story. The origin pushes everything away; the two single-species points (3, 0) and (0, 2) each pull nearby trajectories in. The saddle at (1, 1) is the referee: its two stable directions form a curve — the separatrix — that splits the first quadrant into two basins. Start above the separatrix and you drift to (0, 2); start below it and you land at (3, 0). Connecting the region arrows into smooth non-crossing curves, the whole biological tale appears: with this competition, one species always wins, and which one depends on where you began.

Honest Limits: When the Linear Sketch Can Lie

The classify-by-Jacobian step rests on a real theorem with real fine print. The [[hartman-grobman-theorem|Hartman-Grobman theorem]] guarantees that near an equilibrium the nonlinear flow looks just like its linearization — *provided* the equilibrium is hyperbolic, meaning no eigenvalue has zero real part. For hyperbolic nodes, saddles, and spirals, your linear sketch is trustworthy. The danger case is the [[center|center]]: when the linearization predicts a center (pure imaginary eigenvalues, trace exactly zero), the nonlinear terms you threw away can tip it either way — into a slow inward spiral, a slow outward spiral, or a true center. The linear sketch genuinely cannot decide.

Two more honest reminders. First, the Jacobian tells you only the *local* picture; it says nothing about features that live in the large, such as a closed orbit (a limit cycle) encircling a spiral far from the origin. Global structure has to be argued separately. Second, linearization classifies but does not orient — a linear saddle and a linear node both come with eigenvector directions, yet only the *signs* of the eigenvalues tell you inflow versus outflow. Always carry the sign information through; a beautifully drawn node with the arrows reversed is simply wrong.

Habits That Make Sketches Trustworthy

A few discipline points separate a reliable phase portrait from a hopeful doodle. Trajectories never cross. Because the system is autonomous, the velocity at each point is single-valued, so two distinct trajectories cannot pass through the same point — if your curves cross anywhere except at an equilibrium, you have made an error. Equilibria are the only places trajectories can begin or end in finite reach; elsewhere every curve keeps moving. And arrows on the curves matter as much as the curves: a saddle's stable and unstable directions look identical undecorated, and only the arrowheads distinguish them.

Lean on consistency checks rather than precision. Your trace-determinant classification, your nullcline arrows, and your eigenvector directions all describe the *same* flow, so they must agree — if a point classifies as a stable node but the surrounding region arrows point outward, recheck a sign somewhere. Don't sweat exact curvature or scale; a phase portrait is a *qualitative* object, and the questions it answers (which equilibrium wins, is it stable, does it spiral or oscillate) survive any honest stretching of the picture. Sketch the structure correctly and the geometry can be loose.

That is the whole rung in one motion. Equilibria give the skeleton, nullclines and test points give the global flow, the Jacobian and the trace-determinant plane name each local type, and you stitch the locals into a global story — honestly flagging centers and other borderline cases as open questions. You can now look at almost any planar autonomous system and, without solving a single equation, say where it settles, what oscillates, and what it does in the long run. That qualitative fluency is exactly what the next rungs on nonlinear stability and bifurcations build upon.