Linear Stability Analysis and the Jacobian

Almost no nonlinear system can be solved. The pendulum needs elliptic functions; three gravitating bodies defeated two centuries of analysts; a generic model of two competing species has no closed form at all. Yet dynamicists routinely sketch accurate phase portraits of all of these in minutes. The trick is a strategic retreat: stop solving the flow everywhere and interrogate it only where it stalls — at the fixed points. Near an equilibrium the nonlinear terms are quadratically small, and what remains is governed by a single matrix: the Jacobian, the first derivatives of the velocity field frozen at the fixed point. Linear systems we can solve completely. The Jacobian's eigenvalues hand over the local dynamics on a plate: real parts dictate growth or decay, imaginary parts rotation, eigenvectors the invariant rails along which perturbations travel without turning. One matrix computation replaces an unsolvable differential equation — and, stitched together fixed point by fixed point, these local snapshots organize the entire global portrait. The deep question is whether the approximation can be trusted: does the honest nonlinear flow behave like its linear caricature? The Hartman–Grobman theorem gives the license and its fine print. If the fixed point is hyperbolic — no eigenvalue on the imaginary axis — the nonlinear flow nearby is a continuous distortion of the linear one, same topology, same stability verdict. If an eigenvalue touches the axis, the license is void: the linearization can lie outright, and this article works an explicit system where it does. Those marginal cases are the doorways through which bifurcations enter, and the rest of this chapter walks through them.

Key takeaways

Near a fixed point $\mathbf{x}^*$ of $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ , Taylor expansion leaves $\dot{\boldsymbol{\xi}}=J\boldsymbol{\xi}$ , with $J=D\mathbf{f}(\mathbf{x}^*)$ the Jacobian — a local linear model of an unsolvable flow.
Each eigenpair $(\lambda_k,\mathbf{v}_k)$ is an invariant rail: a perturbation along $\mathbf{v}_k$ evolves as $e^{\lambda_k t}\mathbf{v}_k$ — growth rate $\mathrm{Re}\,\lambda_k$ , rotation $\mathrm{Im}\,\lambda_k$ .
Asymptotic stability is decided by one number: $\max_k \mathrm{Re}\,\lambda_k < 0$ means every perturbation decays; a single eigenvalue with positive real part means escape.
Fast modes die first, so trajectories approach a stable equilibrium tangent to the slowest eigendirection — the geometry behind every "dominant mode" argument.
Hartman–Grobman certifies the linear picture exactly when the fixed point is hyperbolic ( $\mathrm{Re}\,\lambda_k\neq0$ for all $k$ ); on the imaginary axis the certificate is void and cubic terms can overturn the verdict.
Non-hyperbolic fixed points ( $\lambda=0$ or $\pm i\omega$ ) are structurally fragile and are precisely where bifurcations happen — the rest of this chapter's subject.
Key idea: Near a fixed point the nonlinear flow looks like its linearization $\dot{\boldsymbol{\xi}}=J\boldsymbol{\xi}$ ; the eigenvalues of the Jacobian classify stability and local topology, and the picture is faithful whenever the equilibrium is hyperbolic.

Where this lives today

Ask a supercomputer what it is doing at 3 a.m. and odds are it is evaluating Jacobians: stiff-chemistry and circuit solvers Newton-iterate with one at every time step, machine-learning frameworks are industrial Jacobian factories, and the stability of trained recurrent networks — literally flows $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ — is audited through the spectrum of exactly the matrix in equation (2). The stability question has planetary stakes now. Grid operators watch complex eigenvalue pairs drift toward the imaginary axis to forecast inter-area oscillations; climate scientists frame tipping elements (the Atlantic overturning circulation, ice-sheet grounding lines) as equilibria whose leading eigenvalue creeps toward zero, with "critical slowing down" — perturbations recovering ever more slowly, $1/|\sigma_{\max}|\to\infty$ — as the measurable early warning. And spacecraft fly along the machinery itself: the stable and unstable rails of saddle-type Lagrange-point orbits, computed from Jacobians, are the low-fuel highways that took JWST to L2 and keep it there on centimetres-per-second station-keeping.

The physical picture

A marble at the bottom of a bowl returns from any small push — every direction restores. Balanced on a mountain pass, the verdict is mixed: along the ridge the ground pushes back, across it the slightest lean is amplified. Bowl and pass differ not in their fixed point but in the first derivatives of the force field around it — the Jacobian, whose eigenvalues quantify which directions restore, which eject, and how fast.

Figure

The Jacobian is a local microscope — zooming in on a fixed point of a curved nonlinear flow reveals a straight-ruled linear portrait with eigenvector rails

The Jacobian is a microscope: under magnification every smooth field looks flat, and zooming into the fixed point leaves only the linear tilt. The eigenvectors of that tilt are the directions in which the field pushes a perturbation along itself — rails on which displacement scales by $e^{\lambda t}$ without turning. A generic perturbation is a mixture of rail components, each minding its own exponential. The independence of modes shows in every portrait of a stable equilibrium: the fast component collapses almost immediately, leaving the trajectory to creep in along the slow direction. All roads into a node are ultimately tangent to the slowest rail — the engineers' "dominant mode"; Derivation B makes the geometry exact.

Figure

Decay is aristocratic — near a stable node the fast eigendirection empties first and every trajectory arrives tangent to the slow rail

Complex eigenvalues change the choreography: there is no real rail to hug, because the field continuously rotates every perturbation. The real part still budgets growth against decay — a tightening spiral for $\mathrm{Re}\,\lambda<0$ , an unwinding one for $\mathrm{Re}\,\lambda>0$ — while the imaginary part sets the angular speed of the swirl. And exactly on the border, $\mathrm{Re}\,\lambda=0$ , the linear model predicts perfect closed circles forever. That prediction is the one you must not trust. Hyperbolicity — all eigenvalues safely off the imaginary axis — is the condition under which the microscope tells the truth; on the axis, invisible cubic terms take command, and the figure below shows them quietly winding a "center" into a spiral. Derivation C works out the algebra behind these three portraits.

Figure

Three fates behind one linearization — the same linear center is secretly an inward spiral, a true center, or an outward spiral depending on cubic terms the Jacobian cannot see

The core equations

Everything begins with an autonomous flow and a point where it stalls. Suppose the state $\mathbf{x}\in\mathbb{R}^n$ — for the mechanics of this chapter, usually the phase-plane pair $(x,y)=(x,\dot{x})$ — evolves under a smooth velocity field:

\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}),\qquad \mathbf{f}(\mathbf{x}^*)=\mathbf{0} \qquad (1)

where: $\mathbf{f}$ — the vector field assigning a velocity to each state; $\mathbf{x}^*$ — a fixed point, located by solving $\mathbf{f}=\mathbf{0}$ as in the previous article. The physics we are after is what happens near it. The local model is built from the first derivatives of the field, frozen at the equilibrium:

J_{ij}=\left.\frac{\partial f_i}{\partial x_j}\right|_{\mathbf{x}=\mathbf{x}^*} \qquad (2)

where: $J$ — the $n\times n$ Jacobian matrix; entry $J_{ij}$ measures how the $i$ -th velocity component responds to a displacement along the $j$ -th coordinate. For a mechanical system $\dot{x}=y,\ \dot{y}=g(x,y)$ the top row is always $(0,1)$ and the bottom row carries the physics — the stiffness $\partial g/\partial x$ and the damping $\partial g/\partial y$ . Writing $\boldsymbol{\xi}=\mathbf{x}-\mathbf{x}^*$ for the displacement and Taylor-expanding $\mathbf{f}$ about $\mathbf{x}^*$ (Derivation A), the constant term dies by the definition of a fixed point and the quadratic terms are negligible for small $\boldsymbol{\xi}$ :

\dot{\boldsymbol{\xi}}=J\,\boldsymbol{\xi}+\mathcal{O}(|\boldsymbol{\xi}|^2) \qquad (3)

where: $\mathcal{O}(|\boldsymbol{\xi}|^2)$ — the discarded nonlinearity, smaller than the retained term by one factor of the (small) displacement. This is the linearization: an unsolvable flow replaced, in a neighbourhood, by a constant-coefficient linear system — the one class of dynamical systems we can solve in full generality. The general solution is an exponential of a matrix, and its anatomy is clearest in the eigenbasis. If $J$ has eigenpairs $J\mathbf{v}_k=\lambda_k\mathbf{v}_k$ with independent eigenvectors, then

\boldsymbol{\xi}(t)=e^{Jt}\,\boldsymbol{\xi}(0)=\sum_{k=1}^{n} c_k\,\mathbf{v}_k\,e^{\lambda_k t} \qquad (4)

where: $c_k$ — coordinates of the initial perturbation in the eigenbasis; $\mathbf{v}_k$ — the invariant rails of the hero figure. Seed the system on a rail and it stays there; anywhere else, each rail component evolves independently. (A defective $J$ adds secular factors $t\,e^{\lambda t}$ ; the polynomial never beats the exponential, so the verdicts below survive.) Each eigenvalue splits into two physically distinct pieces:

\lambda=\sigma+i\omega:\qquad |e^{\lambda t}|=e^{\sigma t},\qquad \arg e^{\lambda t}=\omega t \qquad (5)

where: $\sigma=\mathrm{Re}\,\lambda$ — the growth rate, the only part controlling magnitude; $\omega=\mathrm{Im}\,\lambda$ — the rotation frequency, swirling without feeding or draining. Conjugate pairs $\sigma\pm i\omega$ describe real spirals with decay time $1/|\sigma|$ and period $2\pi/\omega$ . Stability is now a one-line verdict (proved in Derivation A):

\mathbf{x}^*\ \text{asymptotically stable}\iff \max_k\,\mathrm{Re}\,\lambda_k<0 \qquad (6)

where: the maximum is over all eigenvalues of $J$ . Every mode must decay; a single $\mathrm{Re}\,\lambda>0$ gives generic perturbations an exponentially growing component and the equilibrium is unstable, however fiercely the other modes contract. In the plane — the home terrain of this chapter — the two eigenvalues are captured by two matrix invariants, the trace and the determinant:

\det(J-\lambda I)=\lambda^2-\tau\lambda+\Delta=0,\qquad \lambda_{1,2}=\frac{\tau\pm\sqrt{\tau^2-4\Delta}}{2} \qquad (7)

where: $\tau=\mathrm{tr}\,J=\lambda_1+\lambda_2$ and $\Delta=\det J=\lambda_1\lambda_2$ . Two numbers readable by inspection decide real-versus-complex (sign of $\tau^2-4\Delta$ ), stability (sign of $\tau$ when $\Delta>0$ ), and saddlehood ( $\Delta<0$ ). The full geography of the $(\tau,\Delta)$ plane is the next article, Classification of Fixed Points: Nodes, Saddles, Spirals, Centers; here we need only the stability reading. One flag now: conservative systems pin themselves to $\tau=0$ exactly (the incompressible phase fluid of Conservative Systems and Hamiltonian Phase Flow), forbidding sinks and sources outright — Fixed Points and Stability in Phase Space develops that Hamiltonian case in depth. The license to believe any of this about the original nonlinear system is a theorem:

\text{hyperbolic}\ \big(\mathrm{Re}\,\lambda_k\neq0\ \ \forall k\big)\ \Longrightarrow\ \text{nonlinear flow}\ \underset{\text{loc}}{\cong}\ \text{linear flow} \qquad (8)

where: $\underset{\text{loc}}{\cong}$ — topological conjugacy near $\mathbf{x}^*$ : Hartman–Grobman guarantees a continuous, invertible coordinate change mapping nonlinear trajectories onto linear ones, preserving orientation, time direction, and hence stability type. Hyperbolicity is also structural insurance: eigenvalues off the axis cannot jump across it under a small model change, so the portrait survives imperfect knowledge of $\mathbf{f}$ . The theorem's contrapositive is the map of everything that can go wrong — and of everything interesting to come:

\lambda=\pm i\omega\ \ (\text{linear center})\quad\text{or}\quad \lambda=0\ \ (\det J=0) \qquad (9)

where: these non-hyperbolic cases void the conjugacy. A linear center may be a true center (if energy or reversibility protects it) or a slow spiral either way — Derivation C exhibits the failure. A zero eigenvalue signals a degenerate direction ruled by quadratic and cubic terms — the staging ground of the saddle-node, transcritical, and pitchfork bifurcations ahead.

You want	Reach for	Because
stability verdict	$\max\mathrm{Re}\,\lambda$ of $J$ , eq. (6)	one number decides
approach timescales	$\sigma,\ \omega$ from eq. (5)	settle in $1/\|\sigma\|$ , ring at $2\pi/\omega$
arrival/escape direction	slowest/fastest eigenvectors	leave fast, arrive slow (Derivation B)
quick planar diagnosis	$\tau,\ \Delta$ via eq. (7)	two invariants read by inspection
license to trust it	hyperbolicity, eq. (8)	Hartman–Grobman iff no $\mathrm{Re}\,\lambda=0$
a bifurcation hunt	$\det J=0$ or $\tau=0$ loci, eq. (9)	change requires crossing the non-hyperbolic set

Derivation AFrom the Taylor expansion to the eigenvalue criterion for stability

We derive the linearization honestly from smoothness of the field and push it to the complete stability verdict, equation (6). The strategy is linear algebra's divide-and-conquer: expand, discard what the fixed point kills, rotate into the eigenbasis where the matrix acts as $n$ independent scalars, and bound the full solution by its worst mode. The one step deserving suspicion — discarding the quadratic remainder — is justified not by algebra but by Hartman–Grobman, so watch where it enters.

Step 1. Displace from equilibrium, $\mathbf{x}=\mathbf{x}^*+\boldsymbol{\xi}$ , and Taylor-expand each component of the field about $\mathbf{x}^*$ :

f_i(\mathbf{x}^*+\boldsymbol{\xi})=f_i(\mathbf{x}^*)+\sum_j \left.\frac{\partial f_i}{\partial x_j}\right|_{\mathbf{x}^*}\xi_j+\frac{1}{2}\sum_{jk}\left.\frac{\partial^2 f_i}{\partial x_j\partial x_k}\right|_{\mathbf{x}^*}\xi_j\xi_k+\cdots

Step 2. The constant term vanishes because $\mathbf{x}^*$ is a fixed point, $f_i(\mathbf{x}^*)=0$ ; and since $\mathbf{x}^*$ is constant, $\dot{\boldsymbol{\xi}}=\dot{\mathbf{x}}$ . Keeping the linear term only:

\dot{\xi}_i=\sum_j J_{ij}\,\xi_j+\mathcal{O}(|\boldsymbol{\xi}|^2)

Why the truncation is legitimate: the neglected term is bounded by $C|\boldsymbol{\xi}|^2$ , one power below the retained one. For a hyperbolic point Hartman–Grobman upgrades "small correction" to "topologically irrelevant"; for a non-hyperbolic one the truncation genuinely loses the physics — Derivation C exhibits the failure.

Step 3. Assume $J$ diagonalizable and rotate into the eigenbasis: with $J=P\,\mathrm{diag}(\lambda_1,\dots,\lambda_n)\,P^{-1}$ , define $\boldsymbol{\eta}=P^{-1}\boldsymbol{\xi}$ . The system decouples into scalar equations:

\dot{\eta}_k=\lambda_k\,\eta_k\quad\Longrightarrow\quad \eta_k(t)=\eta_k(0)\,e^{\lambda_k t}

Why the eigenbasis: each $\eta_k$ is the perturbation's coefficient along rail $\mathbf{v}_k$ , where the matrix acts as a mere number — the coupling in $J$ was an artifact of laboratory coordinates.

Step 4. Take magnitudes. Writing $\lambda_k=\sigma_k+i\omega_k$ ,

|\eta_k(t)|=|\eta_k(0)|\,e^{\sigma_k t}

only the real part moves the modulus; the imaginary part spins the phase at rate $\omega_k$ and neither feeds nor drains the mode.

Step 5. Reassemble and bound by the worst offender. With $\sigma_{\max}=\max_k\sigma_k$ ,

|\boldsymbol{\xi}(t)|\;\le\;C\,e^{\sigma_{\max}t}\,|\boldsymbol{\xi}(0)|

for a constant $C$ set by the (fixed) change of basis. If $\sigma_{\max}<0$ every perturbation decays exponentially — asymptotic stability. If some $\sigma_k>0$ , a generic perturbation has $\eta_k(0)\neq0$ and grows as $e^{\sigma_k t}$ — instability. If $J$ is defective the diagonalization is replaced by a Jordan form and solutions acquire polynomial prefactors $t^m e^{\lambda t}$ ; since $t^m e^{\sigma t}\to0$ for any $\sigma<0$ , the criterion (6) is unchanged. Verified: the damped oscillator $\ddot{x}+2\gamma\dot{x}+\omega_0^2x=0$ gives $\lambda=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}$ : underdamped ( $\gamma<\omega_0$ , complex pair), overdamped ( $\gamma>\omega_0$ ), critical ( $\gamma=\omega_0$ , defective root with $t\,e^{-\gamma t}$ ) — the classical damping taxonomy mode by mode. Matches Strogatz §5.1–5.2.

Go deeper · Derivation B · 1 minWhy trajectories arrive along the slow eigendirection

Derivation BWhy trajectories arrive along the slow eigendirection

A stable node has two decaying rails, yet every textbook portrait shows trajectories hugging one. We prove the geometry: generic orbits enter tangent to the eigendirection of smaller $|\lambda|$ — the slow rail. The computation is two lines on the explicit solution; the payoff is the "dominant mode" reasoning used from control engineering to chemical kinetics.

Step 1. Take a planar stable node with real eigenvalues $\lambda_1,\lambda_2$ ordered $-|\lambda_2|<\ \lambda_2<\lambda_1<0$ — so mode 1 is slow, mode 2 fast — and eigencoordinates $(\eta_1,\eta_2)$ along $(\mathbf{v}_1,\mathbf{v}_2)$ :

\eta_1(t)=\eta_1(0)\,e^{\lambda_1 t},\qquad \eta_2(t)=\eta_2(0)\,e^{\lambda_2 t}

Step 2. Form the ratio that measures the direction of the incoming trajectory:

\frac{\eta_2(t)}{\eta_1(t)}=\frac{\eta_2(0)}{\eta_1(0)}\,e^{(\lambda_2-\lambda_1)t}\;\xrightarrow[t\to\infty]{}\;0

since $\lambda_2-\lambda_1<0$ . The fast component does not merely decay — it decays relative to the slow one, so the trajectory's direction rotates onto $\mathbf{v}_1$ . Why this is the whole proof: tangency at the origin is exactly the statement that the transverse coordinate vanishes faster than the longitudinal one, which is what the ratio computes.

Step 3. Eliminate time to see the approach curves. From the two exponentials, $t=\lambda_1^{-1}\ln(\eta_1/\eta_1(0))$ , hence

\eta_2=\text{const}\times \eta_1^{\,\lambda_2/\lambda_1}

with exponent $\lambda_2/\lambda_1>1$ . The orbits are generalized parabolas flattening onto the slow axis — for $\lambda_2/\lambda_1=4$ , quartics $\eta_2\propto\eta_1^4$ , the curves drawn in the second figure.

Step 4. Exceptions and the time-reversed reading: orbits seeded exactly on the fast rail ( $\eta_1(0)=0$ ) come in along $\mathbf{v}_2$ — a measure-zero set. Time reversal swaps the roles, and a saddle inherits both behaviours at once — approach along the stable rail, escape along the unstable one — which is why saddle eigenvectors become the separatrices organizing global portraits. Verified: for $\lambda_1=-1,\ \lambda_2=-4$ , at $t=1$ the fast/slow ratio has shrunk by $e^{-3}\approx0.050$ — an initially diagonal perturbation is within $2.9^\circ$ of the slow rail after one slow time constant; the exponent $\lambda_2/\lambda_1=4$ gives the figure's $\eta_2\propto\eta_1^4$ curves. Matches Strogatz §5.2.

Go deeper · Derivation C · 2 minWhen linearization lies — a center that is secretly a spiral

Derivation CWhen linearization lies — a center that is secretly a spiral

Hartman–Grobman's fine print says non-hyperbolic verdicts cannot be trusted. Here is the canonical counterexample, worked completely: a planar system whose linearization at the origin is a perfect center, but whose true dynamics spirals — inward or outward at our choosing — under an arbitrarily weak cubic term. The strategy: abandon Cartesian coordinates, where the failure is invisible, for polar ones, where the radial dynamics splits off and solves exactly.

Step 1. Consider, for a real parameter $\mu$ ,

\dot{x}=-y+\mu\,x\,(x^2+y^2),\qquad \dot{y}=x+\mu\,y\,(x^2+y^2)

The Jacobian at the origin is computed by dropping the cubic terms: $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ , with eigenvalues $\lambda=\pm i$ — a linear center, independent of $\mu$ . Whatever $\mu$ does, the microscope cannot see it.

Step 2. Pass to polar coordinates $x=r\cos\theta,\ y=r\sin\theta$ . Using $r\dot{r}=x\dot{x}+y\dot{y}$ and $r^2\dot{\theta}=x\dot{y}-y\dot{x}$ :

r\dot{r}=x(-y+\mu xr^2)+y(x+\mu yr^2)=\mu r^4\quad\Longrightarrow\quad \dot{r}=\mu r^3

r^2\dot{\theta}=x(x+\mu yr^2)-y(-y+\mu xr^2)=r^2\quad\Longrightarrow\quad \dot{\theta}=1

Why polar coordinates crack it: the rotation and the radial drift are physically independent motions that Cartesian coordinates entangle; in $(r,\theta)$ the cubic term is revealed as a pure radial pump, invisible at linear order because $r^3\ll r$ near the origin.

Step 3. Solve the radial equation exactly. Separating variables, $\mathrm{d}(r^{-2})/\mathrm{d}t=-2\mu$ , so

r(t)=\frac{r_0}{\sqrt{1-2\mu r_0^2\,t}}

For $\mu<0$ : $r(t)\to0$ — every trajectory spirals into the origin, which is asymptotically stable (but only algebraically fast, $r\sim(2|\mu|t)^{-1/2}$ , far slower than any exponential — another tell that no hyperbolic linear model can mimic it). For $\mu>0$ : $r$ grows and even blows up in finite time $t^*=1/(2\mu r_0^2)$ — the origin is unstable. For $\mu=0$ : $r$ is constant and the center is genuine.

Step 4. The moral: one system, one linearization, three fates. The verdict "center" was not wrong about the linear system — it was wrong to be consulted: with $\mathrm{Re}\,\lambda=0$ Hartman–Grobman is silent and stability is decided at cubic order. Marginal cases need nonlinear tools — a conserved quantity or reversibility to certify a true center (the route of Conservative Systems and Hamiltonian Phase Flow), or a Lyapunov function, here $V=r^2$ with $\dot{V}=2\mu r^4$ . Verified: at $\mu=0$ the system is $\dot{x}=-y,\ \dot{y}=x$ , whose orbits are exact circles ( $\dot{r}=0$ , $\dot{\theta}=1$ ), matching the linear prediction — the failure is strictly a $\mu\neq0$ , cubic-order effect. The example and its polar reduction follow Strogatz §6.3 (Example 6.3.2); rotation sense counterclockwise since $\dot{\theta}=+1$ .

Limiting cases

Case	Result	Meaning
All $\mathrm{Re}\,\lambda_k<0$	asymptotically stable; settling time $\sim1/\|\sigma_{\max}\|$	Every mode decays; hyperbolic, so the verdict is certified.
Some $\mathrm{Re}\,\lambda_k>0$ , some $<0$	saddle-type instability	Generic perturbations escape; stable rails survive as the stable manifold.
All $\mathrm{Re}\,\lambda_k>0$	source; escape rate set by the smallest $\mathrm{Re}\,\lambda$	Time-reverse of a sink: departure is tangent to the slow unstable direction (Derivation B reversed).
Widely split reals, $\|\lambda_2\|\gg\|\lambda_1\|$	fast transient onto a slow manifold	Collapse in time $1/\|\lambda_2\|$ onto the slow rail, then crawl; the workhorse of singular perturbation theory.
$\lambda=\pm i\omega$ (pure imaginary)	non-hyperbolic; linear center, verdict suspended	Nonlinear terms decide (Derivation C); genuine centers need a protecting structure — energy or reversibility.
$\det J=0$ (a zero eigenvalue)	non-hyperbolic; degenerate direction	Equilibria can merge or swap stability along the null direction — the bifurcation gateway.
Hamiltonian flow, any equilibrium	$\tau=0$ forced; $\lambda$ in $\pm$ pairs	Incompressible phase fluid forbids sinks and sources entirely — see Fixed Points and Stability in Phase Space for the conservative story.

Common trapA linear center is not a nonlinear center

The most common misuse of the Jacobian: computing $\lambda=\pm i\omega$ and announcing "closed orbits". Purely imaginary eigenvalues make the fixed point non-hyperbolic — exactly the case Hartman–Grobman excludes — and the true behaviour is decided by the cubic terms the linearization threw away. Derivation C exhibits one system where the same linear center is a stable spiral, a true center, or an unstable spiral depending on an invisible parameter. "Center" is only safe when something beyond linear algebra protects it: a conserved energy, time-reversal symmetry, or an explicit first integral. Absent such structure, treat every linear center as "stability unknown, nonlinear analysis required." The same caution applies to $\lambda=0$ : a vanishing determinant makes the fixed point's very existence fragile, not just its stability.

Applications

Pendulum equilibria, quantitatively. For $\ddot{\theta}+(g/L)\sin\theta=0$ the hanging position gives $\lambda=\pm i\sqrt{g/L}$ — the small-oscillation frequency out of pure linear algebra. The inverted position gives $\lambda=\pm\sqrt{g/L}$ : for a metre stick, $3.1\ \mathrm{s^{-1}}$ , a 0.32 s e-folding time — why a fingertip can just save a metre stick and never a toothpick. (The hanging case is a linear center: it earns "true center" only through energy conservation, per the warning above.)

Control theory is applied eigenvalue-placement. A hovering drone, an inverted pendulum on a cart, a rocket landing on its tail — each is an unstable fixed point with an eigenvalue in the right half-plane. Feedback redesigns the effective Jacobian: measure $\boldsymbol{\xi}$ , actuate $-K\boldsymbol{\xi}$ , and choose the gain $K$ to drag every eigenvalue of $J-K$ into the left half-plane ("pole placement"). Engineering stability margins are literally distances of eigenvalues from the imaginary axis.

Ecology and coexistence. The Jacobian at a coexistence equilibrium decides whether an ecosystem recovers from harvests or outbreaks. Worked Example 2 runs the census on a two-species competition model: coexistence is a saddle — the signature of competitive exclusion, its stable rails the watershed between monoculture fates.

Bifurcation hunting. Qualitative change requires an eigenvalue on the imaginary axis, so parameter studies track $\det J\to0$ (equilibria collide: saddle-node country) and $\tau\to0$ at $\Delta>0$ (a complex pair crosses: onset of oscillation, the Hopf mechanism). Numerical continuation packages follow equilibrium branches while monitoring exactly this spectrum, and every "tipping point" warning about climate or power grids is a statement that some system's $\sigma_{\max}$ is creeping toward zero.

Worked examples

Example 1 — damped pendulum near the bottom (confidence builder). A pendulum with $\ddot{\theta}+b\,\dot{\theta}+\omega_0^2\sin\theta=0$ , $b=0.50\ \mathrm{s^{-1}}$ , $\omega_0=1.0\ \mathrm{s^{-1}}$ , hangs near $\theta=0$ . Classify the equilibrium and find the settling time and ringing period.

Step 1. Write the system: $\dot{\theta}=y$ , $\dot{y}=-\omega_0^2\sin\theta-b\,y$ . The fixed point is $(\theta,y)=(0,0)$ .

Step 2. Jacobian at the origin: $\partial\dot{\theta}/\partial\theta=0$ , $\partial\dot{\theta}/\partial y=1$ , $\partial\dot{y}/\partial\theta=-\omega_0^2\cos\theta\,\big|_0=-1$ , $\partial\dot{y}/\partial y=-b=-0.5$ :

J=\begin{pmatrix}0&1\\-1&-0.5\end{pmatrix},\qquad \tau=-0.5,\quad \Delta=1

Step 3. Discriminant: $\tau^2-4\Delta=0.25-4=-3.75<0$ — complex pair, and $\tau<0$ — decaying. Eigenvalues:

\lambda=\frac{-0.5\pm i\sqrt{3.75}}{2}=-0.25\pm0.968\,i\ \ \mathrm{s^{-1}}

Step 4. Physics from $\sigma$ and $\omega$ : decay time $1/|\sigma|=1/0.25=4.0$ s; ringing period $2\pi/\omega=2\pi/0.968=6.49$ s — a moderately underdamped spiral, about $0.6$ turns per e-folding.

Step 5. Trust check: $\mathrm{Re}\,\lambda=-0.25\neq0$ — hyperbolic, so the verdict holds for the true nonlinear pendulum. The hanging state is an asymptotically stable spiral: e-folding time 4.0 s, ringing period 6.5 s. (Rotation sense: at $(\theta,y)=(1,0)$ the velocity $(0,-\sin 1)$ points down — clockwise, as every oscillator portrait must be.)

Example 2 — competitive exclusion in a two-species model (exam level). Two species share a resource: $\dot{x}=x(3-x-2y)$ , $\dot{y}=y(2-x-y)$ , with $x,y\ge0$ in units of carrying capacity (Strogatz's "rabbits vs sheep"). Find and classify every equilibrium; then say what the phase portrait implies ecologically.

Step 1. Fixed points: set both rates to zero. From the factors: $(0,0)$ ; $(0,2)$ ( $y$ -monoculture); $(3,0)$ ( $x$ -monoculture); and the interior solution of $x+2y=3,\ x+y=2$ , i.e. $(1,1)$ — coexistence.

Step 2. General Jacobian:

J(x,y)=\begin{pmatrix}3-2x-2y & -2x\\ -y & 2-x-2y\end{pmatrix}

Step 3. Boundary equilibria: at $(0,0)$ , $J=\mathrm{diag}(3,2)$ — unstable node. At $(0,2)$ : $J=\begin{pmatrix}-1&0\\-2&-2\end{pmatrix}$ , triangular, $\lambda=-1,-2$ — stable node. At $(3,0)$ : $J=\begin{pmatrix}-3&-6\\0&-1\end{pmatrix}$ , $\lambda=-3,-1$ — stable node.

Step 4. Classify coexistence. At $(1,1)$ :

J=\begin{pmatrix}-1&-2\\-1&-1\end{pmatrix},\qquad \tau=-2,\quad \Delta=(-1)(-1)-(-2)(-1)=1-2=-1<0

Negative determinant — a saddle, no further computation needed. Eigenvalues: $\lambda=\frac{-2\pm\sqrt{4+4}}{2}=-1\pm\sqrt{2}$ , i.e. $\lambda_+=0.414$ (unstable) and $\lambda_-=-2.414$ (stable).

Step 5. Eigenvectors give the watershed: for $\lambda_-=-1-\sqrt2$ , $(J-\lambda_-I)\mathbf{v}=0$ gives $\sqrt2\,v_1-2v_2=0$ , so $\mathbf{v}_-=(\sqrt2,1)$ , slope $+1/\sqrt2\approx0.71$ ; for $\lambda_+$ , $\mathbf{v}_+=(\sqrt2,-1)$ , slope $-0.71$ .

Step 6. Verdict: both monocultures attract; coexistence is a saddle, reachable only along its one-dimensional stable rail. That rail is the separatrix: mixtures on one side drain to all-rabbits $(3,0)$ , on the other to all-sheep $(0,2)$ . Coexistence is dynamically unreachable: generic competition ends in exclusion, the winner decided by which side of the stable manifold history started on. Every eigenvalue has nonzero real part, so the portrait is Hartman–Grobman-certified.

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered

Carl Gustav Jacob Jacobi

1804–1851 · Königsberg

Jacobi never drew a phase portrait — he died half a century before Poincaré invented qualitative dynamics — yet his fingerprints are on every step of this article. The matrix of first derivatives bears his name because his 1841 memoir De determinantibus functionalibus systematized functional determinants as the object controlling how maps stretch volumes and when variables invert — precisely the data linearization freezes at a fixed point. A Königsberg professor of ferocious energy, Jacobi built much of the machinery this chapter's ancestors run on: elliptic functions, which solve exactly the pendulum his successors would rather draw than integrate, and the Hamilton–Jacobi formulation, the deepest classical statement of dynamics as geometry. He worked himself into breakdowns, quarrelled with the Prussian court in 1848, and died of smallpox at 46 — leaving, in the Jacobian, a rare mathematical object every physics student uses weekly without reading a word he wrote.

History

1841

— Jacobi's memoir on functional determinants gives the matrix of first derivatives its lasting role and name

1881

— Poincaré's memoirs on curves defined by differential equations found qualitative dynamics: portraits over formulas

1892

— Lyapunov's thesis distinguishes stability decided by linearization from the marginal cases needing his direct method

1960

— Grobman and Hartman independently prove the conjugacy theorem licensing the linear picture at hyperbolic fixed points

1963

— Lorenz's numerical weather model shows the global consequences local linear analysis cannot see, launching modern chaos theory

Going further

References. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Ch. 5–6 · Taylor, Classical Mechanics, Ch. 11–12 · Goldstein, Poole & Safko, Classical Mechanics, 3rd ed. · Morin, Introduction to Classical Mechanics, Ch. 4 · Landau & Lifshitz, Mechanics, Vol. 1 · Hirsch, Smale & Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos

Provenance. Definitions and the Taylor-expansion derivation follow Strogatz Ch. 5–6 and Hirsch–Smale–Devaney; verifiers should recompute the $2\times2$ characteristic equation $\lambda^2-\tau\lambda+\Delta=0$ and the trace–determinant classification (Strogatz Fig. 5.2.8). The pendulum eigenvalues $\pm i\sqrt{g/L}$ (center) and $\pm\sqrt{g/L}$ (saddle) and the damped-oscillator roots $-\gamma\pm\sqrt{\gamma^2-\omega_0^2}$ are standard textbook values (Taylor Ch. 11–12, Morin Ch. 4). The Hartman–Grobman hyperbolicity statement is standard (Hirsch–Smale–Devaney). Additional worked material in this upgrade: the slow-eigendirection tangency ( $\eta_2\propto\eta_1^{\lambda_2/\lambda_1}$ , Strogatz §5.2), the polar-coordinate counterexample $\dot r=\mu r^3,\ \dot\theta=1$ (Strogatz §6.3), and the competition-model census (Strogatz §6.4, "rabbits vs sheep": interior saddle with $\lambda=-1\pm\sqrt2$ ) — each recomputed here.

Builds on: Fixed Points and Equilibria · Phase Portraits and the Phase Plane · Conservative Systems and Hamiltonian Phase Flow Leads to: Classification of Fixed Points: Nodes, Saddles, Spirals, Centers · Bifurcations and the Loss of Stability · Hopf Bifurcation and Birth of Limit Cycles

If the nonlinear system is unsolvable, how can we ever check that the linearization got the right answer?

Not solution-against-solution — structurally: Hartman–Grobman converts "the discarded terms are small" into "the discarded terms cannot change the topology near a hyperbolic fixed point." Independent cross-checks exist too: conserved quantities and Lyapunov functions give nonlinear verdicts with no expansion, numerics test individual orbits, limiting cases (Derivation C's $\mu=0$ circle check) confirm consistency. When all routes agree, the portrait is as certified as applied mathematics gets.

Is hyperbolicity a rare, special property, or the typical situation?

Typical: failure requires the exact equality $\mathrm{Re}\,\lambda=0$ , a measure-zero condition. But the exceptions matter. Physics manufactures non-hyperbolicity systematically — conservative systems have trace-free Jacobians, so their centers sit permanently on the marginal line. And any system with a tunable parameter crosses the marginal set whenever its behaviour changes qualitatively; those crossings are bifurcations. Hyperbolicity is the generic backdrop; its failures are the plot.

Does the Jacobian have to be evaluated exactly at the fixed point?

For this article's theory, yes — that is where the constant term of the Taylor expansion dies and the linear term leads. Evaluated elsewhere, $J(\mathbf{x})$ measures the local shearing of the flow around a moving point, and its eigenvalues there carry no direct stability meaning. The honest generalizations are Lyapunov exponents, which average the time-varying Jacobian along a trajectory. Numerically, a slightly-off evaluation point is harmless — eigenvalues are continuous in the entries — provided the fixed point itself was located accurately first.

Why keep only the first derivative? Wouldn't quadratic terms give a better approximation?

More accurate, rarely more useful: near a hyperbolic fixed point the linear term dominates and fixes the topology — higher terms bend trajectories without changing the fixed point's kind, which is Hartman–Grobman's content. Quadratic and cubic terms earn their keep exactly when the linear verdict is marginal: at non-hyperbolic points, center-manifold reduction plus normal forms keeps the lowest nonlinear terms that matter. Rule: linear order settles hyperbolic cases completely; the first non-negligible nonlinear order settles the marginal ones.

The damped pendulum's Jacobian at the bottom has complex eigenvalues. What is physically rotating?

Nothing in real space — the rotation is in the phase plane. The state $(\theta,\dot\theta)$ loops the fixed point: swing right while slowing, turn, swing back while speeding up — one swing cycle is one $2\pi$ of phase-plane rotation at rate $\omega=\mathrm{Im}\,\lambda$ . A conjugate pair $\sigma\pm i\omega$ is exactly a real rotation-plus-scaling block: the imaginary part is the ringing frequency you would hear, the real part the rate at which each swing is quieter. Complex numbers are just the plane's best notation for "decay with rotation."

How does this machinery extend beyond two dimensions?

Equations (1)–(6) and (8) hold verbatim for any $n$ : compute the spectrum, $\max\mathrm{Re}\,\lambda$ decides, hyperbolicity licenses. Two things change: the $(\tau,\Delta)$ shortcut is planar-only (higher dimensions need the eigenvalues or Routh–Hurwitz), and globally the options explode — planar trajectories cannot cross (Poincaré–Bendixson, two articles ahead), while in three or more dimensions manifolds can weave into chaos. Local stays easy; global does not.

My Jacobian has a zero eigenvalue. What do I actually do next?

Accept what you cannot conclude: both stability and the equilibrium's very existence are in doubt along the null direction. The systematic tool is center-manifold reduction: an invariant curve tangent to the zero-eigenvalue direction carries all the slow, undecided dynamics while hyperbolic directions decay onto it. Restrict to that manifold, expand, and the lowest surviving nonlinear term gives the verdict — generically the saddle-node, transcritical, or pitchfork normal forms closing this chapter. A zero eigenvalue is not a dead end but a forwarding address.

Are the eigenvalues of the Jacobian the same as Lyapunov exponents?

At a fixed point, essentially yes: the exponents of the constant solution are the eigenvalues' real parts. Elsewhere, no. Lyapunov exponents are long-time average separation rates, computed by transporting a perturbation with the time-varying Jacobian $J(\mathbf{x}(t))$ along an orbit; for periodic orbits they become Floquet multipliers, and a positive largest exponent is the quantitative definition of chaos. This article is the stationary special case: linearize along a motion, not just at a standstill.

Linear Stability Analysis and the Jacobian

Where this lives today

The physical picture

The core equations

Limiting cases

Applications

Worked examples

Going further

Before this

Where this leads