Equilibria and What Happens Near Them

Where the flow stands still

The previous guide taught you to read a planar system x' = f(x, y), y' = g(x, y) as a velocity vector field: at every point an arrow says which way and how fast a particle would drift, and the trajectories are the curves that thread along those arrows. Now ask the simplest possible question about such a flow: where does nothing move? A point where the velocity arrow shrinks to nothing — where both f = 0 and g = 0 at once — is an equilibrium point (also called a critical or fixed point). Drop a particle exactly there and it sits forever; the trajectory is a single motionless dot.

Finding equilibria is therefore pure algebra, not calculus: solve the two equations f(x, y) = 0 and g(x, y) = 0 simultaneously. Geometrically each equation carves out a curve in the plane — the nullcline where one component of the velocity vanishes. On the curve f = 0 the flow is purely vertical (only y moves); on g = 0 it is purely horizontal. An equilibrium lives exactly where these two nullclines cross, because only there does *both* the horizontal and the vertical motion stop at the same time. A system can have one such crossing, several, or none at all.

Standing still is not the interesting part

Knowing *where* the flow stands still is only the opening line. The whole drama is what happens to a particle placed *near* an equilibrium but not exactly on it. Three radically different fates are possible, and they decide the character of the entire portrait. The particle may be pulled steadily inward, so the equilibrium swallows everything around it — a sink. It may be pushed away, so the slightest displacement grows without bound — a source. Or it may circle around without ever quite landing or escaping. Same motionless dot, utterly different neighbourhoods.

This is exactly the notion of stability carried up from one dimension. On the phase line you classified each equilibrium as a sink or source by checking the sign of f' there. In the plane the same questions sharpen into a vocabulary of *shapes*: an equilibrium that draws everything in might do so along straight lines (a node) or by spiralling (a spiral); one that lets some directions in and others out is a saddle; one ringed by closed loops is a center. Naming and telling apart those shapes is the job of the very next guide. This guide answers the prior question: *how* do we read the local behaviour off the equations at all?

Zoom in until the curve looks straight

Here is the central idea of the whole subject, and it is borrowed straight from beginner calculus. Up close, a smooth curve looks like its tangent line; that is the entire meaning of a derivative. The same is true for a flow. Park yourself at an equilibrium, throw away everything except the very first slope of f and g, and the curved nonlinear field is replaced by a flat linear one — a system x' = A x that you already know how to solve completely with eigenvalues. This replacement is called linearization about the equilibrium, and it is what makes the phase plane tractable at all.

Concretely, shift coordinates so the equilibrium sits at the origin, write u and v for the tiny displacement, and keep only the linear terms of f and g. Those linear terms are precisely the partial derivatives of f and g, packed into a 2-by-2 matrix called the Jacobian matrix, evaluated *at the equilibrium*. The Jacobian is the multivariable version of the single slope f' that ran the phase-line story. Its four entries are the four ways the two velocities respond to the two displacements, and they are all you need for the local picture.

Nonlinear:   x' = f(x, y)
             y' = g(x, y)

At an equilibrium (x*, y*):   f(x*, y*) = 0,  g(x*, y*) = 0

Let u = x - x*,  v = y - y*  (small displacement). Keep only linear terms:

     [ u' ]     [ f_x   f_y ] [ u ]
     [    ]  =  [           ] [   ]       all partials evaluated at (x*, y*)
     [ v' ]     [ g_x   g_y ] [ v ]
             \_______________/
                Jacobian  J  =  A

Local behaviour  <-  eigenvalues of J.

Now the payoff. The local system u' = J u is exactly the constant-coefficient linear system from the previous rung, so its fate is decided entirely by the eigenvalues of J. Real negative eigenvalues pull inward (a sink); real positive ones push outward (a source); one of each sign gives a saddle; complex eigenvalues spiral, with the real part setting growth or decay. The eigenvalues you already know how to compute *are* the answer to 'what happens near this equilibrium'. The qualitative phase plane is, at heart, the eigenvalue method read geometrically.

The recipe, step by step

Putting the pieces together gives a short, reliable procedure you will run at every equilibrium of every planar system from now on. It is worth memorizing because the last three guides of this rung all lean on it.

1. Find every equilibrium by solving f(x, y) = 0 and g(x, y) = 0 together — algebra, often where the nullclines cross.
2. At one equilibrium, form the Jacobian J of (f, g) — the four partial derivatives f_x, f_y, g_x, g_y — and evaluate them at that point.
3. Compute the eigenvalues of J from det(J - lambda I) = 0 (a quadratic in lambda).
4. Read off the local type from the eigenvalues: both negative real (sink), both positive (source), opposite signs (saddle), complex (spiral, or center if purely imaginary).
5. Repeat for every equilibrium, then stitch the local pictures together into the global phase portrait — the subject of the final guide.

Notice that you never have to *solve* the nonlinear system to learn its local behaviour — and that is the whole point. Most nonlinear ODEs have no closed-form solution at all, so a method that reads off qualitative fate from a single matrix of derivatives is not a convenience but a necessity. With the eigenvalues in hand, each equilibrium gets a label and a little sketch, and the global phase portrait becomes the patient act of joining those local sketches with consistent flow arrows.

When linearization can lie

Be honest: replacing a curved field by its linear part is an approximation, and approximations have a fine print. The theorem that licenses it is the Hartman-Grobman theorem, and it makes a precise promise: near an equilibrium the true nonlinear flow looks just like its linearization — same node, saddle, or spiral — *provided no eigenvalue of the Jacobian has zero real part*. An equilibrium meeting that condition is called hyperbolic, and at a hyperbolic equilibrium the linear sketch is trustworthy down to the qualitative shape.

The danger sits exactly on the boundary. When an eigenvalue has zero real part — a pure imaginary pair, the linear center case being the classic culprit — the theorem stays silent, and for good reason. The linearization may report neat closed loops, while the discarded higher-order terms quietly turn them into a slow inward spiral, or a slow outward one. Those tiny terms, negligible for fate elsewhere, become the *deciding* factor here precisely because the linear part is balanced on a knife edge. A linear center is the most fragile prediction in the whole theory.