Foundations — Phase plane analysis — trajectories, critical points

This page assumes nothing. If the parent note used a symbol, we build it here from a picture first. Read top to bottom: each block earns the next.

0. The very first picture: a plane full of arrows

Why a plane and not a line? Because our system has two changing quantities, $x$ and $y$. One number needs a line; two numbers need a plane. Each dot is a complete description of "the state at this instant".

Look at figure s01: a single dot is now. As time passes the dot slides across the sheet, dragging a curve behind it. We are about to give names to every ingredient that makes the dot move.

1. $x(t)$ and $y(t)$ — quantities that depend on time

Picture: $x(t)$ is the horizontal coordinate of our moving dot at the moment the clock reads $t$; $y(t)$ is its vertical coordinate. Feed in $t=0,1,2,\dots$ and you get the dot's whereabouts frame by frame.

Why the topic needs it: the whole subject is about change over time. Without a clock $t$, nothing moves and there is no trajectory to draw.

2. The dot $\dot{x}$ — "rate of change" (the derivative, from zero)

Newton's shorthand: a dot over a letter means "how fast that letter changes as time ticks".

WHY the derivative and not just "the change"? A plain change like "$x$ went up by 3" hides how quickly. The derivative captures the change at an instant — the slope of the graph of $x$ against $t$ right now. That instant-by-instant velocity is exactly what an arrow needs to point right now.

Picture (figure s02): at the moving dot, $\dot{x}$ is the length of its rightward push and $\dot{y}$ the length of its upward push. Glue those two pushes tail-to-tail and you get one arrow — the velocity arrow $(\dot{x},\dot{y})$. That arrow is the direction the dot is about to travel.

3. $f(x,y)$ and $g(x,y)$ — the rule that hands you the arrow

The system is written as $\dot{x}=f(x,y),\dot{y}=g(x,y).$

Reading this out loud: "my rightward speed $\dot{x}$ equals whatever the rule $f$ says at my current spot; my upward speed $\dot{y}$ equals whatever the rule $g$ says." So $f$ and $g$ paint the arrow at every point of the plane.

Why the topic needs it: $f$ and $g$ are the wind. Give me any spot $(x,y)$ and these two functions immediately give me the velocity arrow there — no solving required. Doing this at many spots covers the plane with arrows: the direction field.

4. "Autonomous" — why $t$ must be missing from $f$ and $g$

Picture: the wind at each spot is frozen in place — the same every day. Return to a spot tomorrow and the arrow there is identical.

Why this is the whole engine of the subject: because the arrow depends only on position, one direction lives at each point. Two different trajectories could never pass through the same ordinary point going different ways — there is only one arrow to obey. This is why trajectories never cross (except at the special still-points of §6).

5. $(\dot{x},\dot{y})$ as a vector, and slope $dy/dx$

Picture: identical to the velocity arrow of figure s02. The slope $dy/dx$ is just the tilt of that arrow — rise $\dot{y}$ over run $\dot{x}$ — with the clock thrown away.

Why both viewpoints: the vector $(\dot{x},\dot{y})$ keeps speed and direction; the slope $dy/dx$ keeps only direction. The subject uses the vector to see flow with arrows, and the slope to describe the bare curve shapes.

6. Critical point — where the arrow has zero length

Picture (figure s04): a spot of dead calm in the windy field. Drop the leaf exactly there and it never moves. Every other arrow in the plane arranges itself around these still-points, so finding them is like finding the drains and fountains that shape all the flow.

Why "critical": these are the only places a trajectory can start, end, or meet, and they organize the entire portrait. Find them by solving the two equations $f=0$ and $g=0$ simultaneously (ordinary algebra, no calculus).

7. Partial derivatives $f_x,f_y,g_x,g_y$ — slope in one direction at a time

To zoom in near a still-point we need "how does the arrow change as I step a little?" — but in two directions. That is what a partial derivative measures.

Picture: stand on the surface height $f(x,y)$. $f_x$ is the steepness if you walk east (along $x$); $f_y$ is the steepness if you walk north (along $y$). Two separate slopes because you can tilt in two independent directions.

Why the topic needs it: near a critical point the curved wind field looks straight — and the four partials $f_x,f_y,g_x,g_y$ are exactly the four numbers that describe that straight (linear) approximation.

8. The Jacobian $J$ — the four partials packed in a box

Reading it: the top row tells how $\dot{x}$ responds to a nudge in $x$ (left) and in $y$ (right); the bottom row does the same for $\dot{y}$. So $J$ is the local instruction sheet for the arrows in the tiny neighborhood of the still-point.

See Linearization and the Jacobian matrix for the full Taylor-expansion story; here we only need to recognize $J$ as this box of four slopes.

9. $\mathrm{tr}J$ and $\det J$ — two summary numbers of the box

Two single numbers squeeze the essence out of the $2\times 2$ box:

Picture: think of $\det J$ as "how much a tiny square of leaves gets stretched or squashed and flipped" by the local arrows, and $\mathrm{tr}J$ as "how much the leaves puff outward on net". Their signs alone (positive/negative) already decide the type of still-point. See the Trace–determinant plane for the full map.

Why: you rarely need the full box — these two numbers carry the verdict, as §11 shows.

10. $\lambda$, eigenvalues, and $e^{\lambda t}$ — the growth/rotation dials

Why $e^{\lambda t}$ shows up: guess a solution shaped like $\mathbf{v}{e}^{\lambda t}$ (an arrow whose length is scaled by the growth factor $e^{\lambda t}$). Plug in and the equation forces $J\mathbf{v}=\lambda \mathbf{v}$ — so $\lambda$ must be an eigenvalue. See Eigenvalues and eigenvectors.

The two eigenvalues come from ${\lambda }^2-(\mathrm{tr}J)\lambda +\det J=0.$ So the two summary numbers of §9 feed straight into the two dials ${\lambda }_{1,2}$ that decide everything.

11. Putting the dials together — the four portraits

For deeper stability tools beyond linearization, see Stability and Lyapunov functions; for the outward-spiral-to-a-ring behavior, see Limit cycles and Poincaré–Bendixson theorem.

How the foundations feed the topic

Equipment checklist

Test yourself — cover the right side, recall it, then reveal.