Hartman-Grobman: When Linearization Tells the Truth

The promissory note you have been spending

In the previous guide you did something bold and slightly suspicious. Faced with a curvy nonlinear system x' = f(x), you stood at an equilibrium, computed the Jacobian matrix J there, threw away every term of higher order, and declared: *near this point the system behaves like x' = J x.* You then read off the eigenvalues and called the equilibrium a node, a saddle, or a spiral. It worked beautifully on the examples. But a careful reader should be uneasy. You deleted the very nonlinearity that makes the equation interesting — what gives you the right to believe the leftover linear cartoon resembles the real thing at all?

That uneasiness is exactly right, and it deserves an honest answer rather than a shrug. Linearization is a *promissory note*: it claims the local picture of the nonlinear system matches the local picture of its linearization. The note is not free — somebody has to guarantee it can be cashed. The Hartman-Grobman theorem is that guarantee. It states the precise condition under which the swap is legal, and — just as important — it draws a sharp line around the one situation where the note bounces. This guide is about reading that contract carefully, because using linearization without knowing its fine print is how confident students reach confidently wrong conclusions.

What 'behaves like' has to mean

Before we can prove anything, we have to pin down the slippery phrase 'behaves like'. It cannot mean the solutions are *equal* — they obviously are not; the nonlinear trajectories curve where the linear ones go straight. It cannot even mean they stay numerically close forever; a tiny initial gap between a true solution and its linear approximation can grow without bound as time runs on. So what survives? The answer is the *shape of the phase portrait* — the qualitative pattern of how trajectories arrange themselves around the equilibrium: which ones flow in, which flow out, how many directions there are, whether things spiral. That pattern, and not the exact curves, is what we ask to be preserved.

The mathematicians' way to say 'same shape' is topological equivalence, or topological conjugacy. Imagine the phase portrait of the linear system printed on a sheet of perfectly stretchy rubber. You may bend, stretch, and warp that sheet however you like — squashing here, pulling there — provided you never tear it and never glue two points into one. Any picture you can reach by such gentle distortion counts as 'the same'. Two portraits are topologically equivalent when one is just a continuously bent copy of the other, with the direction of flow along each trajectory preserved. This is a generous notion of sameness — it ignores all metric detail like angles, speeds, and exact curvature — and that generosity is precisely why a theorem about it can possibly be true.

The theorem, and its one demand

Now the statement, stripped to its core. Take a smooth system x' = f(x) with an equilibrium at a point we may as well call the origin, and let J be its Jacobian there. The Hartman-Grobman theorem says: *if every eigenvalue of J has nonzero real part, then in a small neighbourhood of the equilibrium the nonlinear flow is topologically equivalent to the flow of the linear system x' = J x.* That single condition — no eigenvalue sitting on the imaginary axis — is the whole price of admission. An equilibrium meeting it is called a hyperbolic equilibrium, and hyperbolicity is the magic word for this entire rung.

Why should a nonzero real part be the thing that matters? Recall from the linear theory that the real part of an eigenvalue controls growth or decay: lambda with positive real part stretches a direction outward like e^(rt) blowing up, negative real part shrinks it in. As long as no eigenvalue has real part exactly zero, every direction has a *definite verdict* — either firmly attracting or firmly repelling — and there is enough 'pull' to overpower the small nonlinear terms nearby. The deleted higher-order pieces are like e^2 next to a sturdy e: real, but too feeble to flip a verdict that the linear part has already delivered decisively. Hyperbolicity is exactly the condition that the linear part speaks with enough authority to drown out the correction.

  eigenvalues of J          equilibrium is...     linearization is...
  --------------------------------------------------------------------
  all Re(lambda) < 0        sink (stable node/spiral)   TRUSTWORTHY
  all Re(lambda) > 0        source (unstable)           TRUSTWORTHY
  mixed signs, none = 0     saddle                      TRUSTWORTHY
  --------------------------------------------------------------------
  some Re(lambda) = 0       NON-hyperbolic              VERDICT WITHHELD
  (e.g. pure imaginary  ->  linear center)              linear picture may LIE

Where the truth runs out: the center

Now the honest, important part — the case the theorem deliberately refuses to cover. Suppose the Jacobian has a pair of purely imaginary eigenvalues, real part exactly zero. The linear system x' = J x then has a center: a nest of perfect closed loops, circling the origin forever, neither winding in nor spiralling out. The equilibrium is *not* hyperbolic, so Hartman-Grobman says nothing — and that silence is not a technicality to wave away. It is a genuine warning, because here the linear picture really can be a lie.

Here is a concrete shock. Consider the system x' = -y + a x(x^2 + y^2), y' = x + a y(x^2 + y^2). Its Jacobian at the origin has eigenvalues +i and -i — a textbook center, closed loops, by the linear reading 'stable but not attracting.' Yet switch to a radius variable r = sqrt(x^2 + y^2) and the true equation collapses to r' = a r^3. If a is even slightly positive, r grows and every trajectory spirals slowly *outward* to infinity — an unstable spiral. If a is slightly negative, r shrinks and everything spirals *inward* — an asymptotically stable spiral. The linear center predicted neither; the discarded cubic term, powerless at a hyperbolic point, is the lone voice in the room when the linear part falls silent on the imaginary axis. Same linearization, opposite fates, decided entirely by the nonlinearity you were tempted to throw away.

Using the licence in practice

Put together, the theorem turns the loose habit of the last guide into a disciplined procedure. The whole point is that the costly question — how does the full nonlinear flow look near here? — gets answered by the cheap question — what are the eigenvalues of one matrix? — but *only after* you have earned the right by checking hyperbolicity. Skip that check and you are no longer doing mathematics, just hoping. Here is the contract, walked through in order.

1. Find every equilibrium: solve f(x) = 0 for all components at once. These are the points where the velocity vanishes — the skeleton of the phase portrait.
2. At each equilibrium compute the Jacobian J — the matrix of partial derivatives evaluated at that point. This is the best linear stand-in for f near there.
3. Find the eigenvalues of J and read their real parts. This is the verdict-gathering step inherited straight from the linear theory.
4. Check hyperbolicity: is every real part nonzero? If yes, the equilibrium is hyperbolic and you have the green light.
5. If hyperbolic, classify with confidence: all real parts negative -> stable sink; all positive -> source; mixed signs -> saddle. Hartman-Grobman guarantees the nonlinear flow matches this local picture.
6. If some real part is zero (non-hyperbolic), STOP. Declare the linear test inconclusive and switch to Lyapunov's method or a center-manifold analysis — the nonlinearity now holds the deciding vote.

One last honesty, often glossed over: even in the best case the guarantee is strictly *local*. Hartman-Grobman speaks only inside some small neighbourhood of the equilibrium whose size it never tells you. It says the saddle near the origin really is a saddle; it says nothing about how far that saddle's influence reaches, where two basins of attraction meet, or what global structures — like a closed orbit far out — might dominate the large-scale flow. Linearization is a microscope, not a map of the whole country. Knowing both what it certifies and where its vision ends is the difference between wielding it and being misled by it.

What you now carry forward

You came into this guide with a useful trick and a nagging doubt. You leave with the trick turned into a theorem and the doubt turned into a precise boundary. Linearization is not a vague approximation you cross your fingers over — it is a rigorous tool with a clearly stamped warranty: valid at hyperbolic equilibria, void on the imaginary axis. That single distinction, hyperbolic versus not, is the hinge the rest of the rung swings on. When the linear part decides, lean on it fully; when it abstains, you already know who to call.

That 'who to call' is the very next guide. The center case we hit head-on here — pure imaginary eigenvalues, the linearization shrugging — is exactly the gap Lyapunov's direct method was built to fill. It will let you certify stability by watching an energy-like quantity drain away, with no eigenvalues, no solving, and no need for the equilibrium to be hyperbolic. Hartman-Grobman drew the map of where linearization works; the next tool is for the territory it left blank.