Phase Portraits and Fixed Points

A phase portrait is a geometric picture of all solution trajectories of an autonomous system of differential equations in its state space, while fixed points (equilibria) are the points where the system does not move. Key result: the local behaviour near a hyperbolic fixed point is completely determined by the eigenvalues of the linearization (Hartman–Grobman). Applies to: dynamical systems, mechanics, ecology, circuits, neuroscience. Who/when: Poincaré (1880s) founded the qualitative theory; Lyapunov, Andronov, Hartman, and Grobman extended it through the 20th century.

A phase portrait displays the qualitative geometry of solutions of an autonomous system $\dot{x} = f (x)$ by drawing representative trajectories as curves in the state space, with arrows indicating the direction of flow. A fixed point $x^{*}$ satisfies $f (x^{*}) = 0$ : a system started there stays there forever. The central reason this matters is that most nonlinear ODEs cannot be solved in closed form, yet their long-term qualitative behaviour — equilibria, stability, oscillations, basins of attraction — can be extracted geometrically without ever writing down a formula for the solution. The surprising fact is that near a "generic" (hyperbolic) equilibrium, the wildly nonlinear flow is topologically identical to that of a simple linear system — so a $2 \times 2$ matrix of partial derivatives often tells you everything about local dynamics.

The Core Insight

Forget solving for $x (t)$ . Instead, treat $f (x)$ as a velocity field painted over state space. Solutions are the streamlines you would trace by releasing a particle and letting it ride that field. The skeleton of the whole picture is the set of fixed points plus the invariant curves (eigendirections, separatrices, limit cycles) that organize how trajectories flow between them. Classify each fixed point, glue the local pictures together, and you have understood the system.

Formal Statement

Autonomous System and Phase Space

An autonomous first-order system in $n$ dimensions is

\dot{x} = f (x), x \in R^{n}, f : R^{n} \to R^{n} .

⟳"Autonomous" means the right-hand side has no explicit time dependence; the dot denotes

d / d t

. Each

x

is a point in

phase space $R^{n}$ .

The flow $φ_{t} (x_{0})$ is the solution at time $t$ with $φ_{0} (x_{0}) = x_{0}$ . The image of $t \mapsto φ_{t} (x_{0})$ is a trajectory (orbit). The phase portrait is the collection of all trajectories with flow directions marked.

Fixed Point

A point $x^{*}$ is a fixed point (equilibrium, critical point, steady state) if

f (x^{*}) = 0 .

⟳At such a point the velocity vanishes, so

φ_{t} (x^{*}) = x^{*}

for all

t

; a constant function is then a solution of [1].

Jacobian and Linearization

Near a fixed point, write $x = x^{*} + u$ with $u$ small. Taylor-expand:

\dot{u} = A u + O (∥ u ∥^{2}), A = D f (x^{*}), A_{ij} = \frac{\partial f _{i}}{\partial x _{j}}_{x^{*}} .

⟳Substitute

x = x^{*} + u

into [1], use

f (x^{*}) = 0

and the multivariable Taylor theorem;

A

is the

Jacobian matrix evaluated at $x^{*}$ .

The matrix $A$ governs the linearized system $\dot{u} = A u$ , whose solution is $u (t) = e^{A t} u (0)$ .

For planar systems ( $n = 2$ ) the classification is exhaustive and visual. Write the eigenvalues of $A$ as $λ_{1, 2}$ , with trace $τ = tr A = λ_{1} + λ_{2}$ and determinant $Δ = det A = λ_{1} λ_{2}$ .

Trace–Determinant Classification (n = 2)

For $\dot{u} = A u$ with $A$ a real $2 \times 2$ matrix having $det A \neq = 0$ (isolated fixed point at origin), the eigenvalues satisfy

λ_{1, 2} = \frac{τ \pm τ ^{2} - 4Δ}{2},

⟳Roots of the characteristic polynomial

λ^{2} - τ λ + Δ = 0

obtained from

det (A - λ I) = 0

and the fixed point type is:

$Δ < 0$ : saddle (real eigenvalues of opposite sign), always unstable.
$Δ > 0, τ^{2} - 4Δ > 0$ : node — stable if $τ < 0$ , unstable if $τ > 0$ .
$Δ > 0, τ^{2} - 4Δ < 0, τ \neq = 0$ : spiral (focus) — stable if $τ < 0$ , unstable if $τ > 0$ .
$Δ > 0, τ = 0$ : center (purely imaginary eigenvalues), neutrally stable (borderline).
$τ^{2} - 4Δ = 0$ : degenerate/improper node (repeated eigenvalue), borderline case.

Hartman–Grobman

If $x^{*}$ is a hyperbolic fixed point — meaning $A = D f (x^{*})$ has no eigenvalue with zero real part — then there is a homeomorphism of a neighbourhood of $x^{*}$ carrying trajectories of the nonlinear flow [1] onto trajectories of the linear flow $\dot{u} = A u$ , preserving the direction of time.

Hartman–Grobman fails precisely at the borderline cases: centers ( $τ = 0$ ) and degenerate nodes ( $τ^{2} = 4Δ$ ) are non-hyperbolic. There, nonlinear terms decide stability and the linearization is unreliable — see the counterexample below.

Why It Holds / Derivation

Linear flow determines local geometry

Step 1. Near

x^{*}

, equation [3] says

\dot{u} \approx A u

. The exact solution of the linear part is

u (t) = e^{A t} u (0)

. Step 2. Diagonalize (or Jordan-decompose)

A

. If

A

has distinct eigenvalues

λ_{1}, λ_{2}

with eigenvectors

v_{1}, v_{2}

, then any initial condition decomposes as

u (0) = c_{1} v_{1} + c_{2} v_{2}

and

u (t) = c_{1} e^{λ_{1} t} v_{1} + c_{2} e^{λ_{2} t} v_{2} .

⟳Apply

e^{A t}

to each eigenvector using

e^{A t} v = e^{λ t} v

, which follows from

A v = λ v

and the series definition of

e^{A t}

Step 3. Read off geometry from the signs/forms of

λ_{i}

Both real negative ⇒ every term decays ⇒ trajectories funnel inward (stable node).
Opposite signs ⇒ one direction grows, one decays ⇒ saddle with the eigenvectors as the stable ( $λ < 0$ ) and unstable ( $λ > 0$ ) manifolds.
Complex $λ = α \pm i β$ ⇒ $e^{α t} (cos β t, sin β t)$ ⇒ rotation with exponential envelope ⇒ spiral if $α \neq = 0$ , closed loop if $α = 0$ (center).

Step 4. The eigendirections are invariant lines of the linear flow: a trajectory starting on

v_{i}

stays on it forever, evolving as

e^{λ_{i} t}

. These straight lines are the organizing skeleton. Step 5. Hartman–Grobman (proof omitted; uses a contraction/fixed-point argument on the space of conjugating maps) guarantees that for hyperbolic

x^{*}

the nonlinear trajectories are continuous deformations of these linear ones. Hence Step 3's classification transfers to the full system [1].

■

Five canonical planar phase portraits: stable node, saddle, stable spiral, center, and unstable node.

Henri Poincaré | 1854–1912

Poincaré single-handedly created the qualitative theory of differential equations in his 1881–1886 memoirs Sur les courbes définies par une équation différentielle. Confronting the three-body problem — which he proved has no closed-form solution — he shifted the question from "what is the formula?" to "what does the family of all solutions look like?". He classified planar singular points (nodes, saddles, foci, centers), introduced the index of a fixed point, the Poincaré map (return map) for periodic orbits, and the index-sum theorem relating equilibria to the topology of the phase space. His geometric viewpoint is the foundation of all modern dynamical systems theory and chaos.

Building Intuition

Imagine a marble on a sculpted landscape, but instead of gravity pulling it downhill, an arrow field tells it which way to roll at each spot.

A stable node is a drain: arrows point straight in from all sides, marbles slide directly to the center.
A saddle is a mountain pass: along one ridge you slide away, along the perpendicular valley you slide toward the pass. Only the special "stable manifold" curve actually reaches it; everything else eventually escapes.
A spiral is a whirlpool: arrows have a rotational twist plus inward (or outward) pull, so marbles circle while they sink (or fly out).
A center is a frictionless racetrack: marbles orbit forever on closed loops, neither gaining nor losing.

The phase portrait is just the union of all these marble-paths. To sketch one by hand: find fixed points, linearize at each, classify by [4], draw the local picture, then connect them respecting the global flow direction (use nullclines $f_{1} = 0$ and $f_{2} = 0$ where the field is purely vertical/horizontal as scaffolding).

Worked Examples

Classifying a linear system

Given:

\overset{x}{˙} = x + 3 y, \overset{y}{˙} = - x - y

. Step 1 — Fixed point. Set both to zero:

x + 3 y = 0

and

x + y = 0

. Subtracting gives

2 y = 0 \Rightarrow y = 0

, then

x = 0

. Unique fixed point at the origin. Step 2 — Jacobian. The system is linear, so

A = (1 - 1 3 - 1) .

Step 3 — Trace and determinant.

τ = 1 + (- 1) = 0

Δ = (1) (- 1) - (3) (- 1) = - 1 + 3 = 2

. Step 4 — Eigenvalues. From [4]:

λ = \frac{0 \pm 0 - 8}{2} = \pm i 2

. Purely imaginary. Step 5 — Classify.

Δ > 0, τ = 0

⇒ center. Trajectories are closed ellipses orbiting the origin (here exactly, since the system is linear).

Center; period T = \frac{2 π}{β} = \frac{2 π}{2} \approx 4.44 (time units) .

Nonlinear: the damped pendulum

Given:

\ddot{θ} + 0.5 \dot{θ} + sin θ = 0

. Write as a system with

x = θ

y = \dot{θ}

\overset{x}{˙} = y, \overset{y}{˙} = - 0.5 y - sin x .

Step 1 — Fixed points. Need

y = 0

and

sin x = 0

⇒

x = nπ

. So equilibria at

(0, 0)

(hanging down) and

(π, 0)

(balanced up), repeating with period

2 π

. Step 2 — Jacobian.

A (x, y) = (0 - cos x 1 - 0.5) .

Step 3 — At $(0, 0)$ :

cos 0 = 1

, so

A = (0 - 1 1 - 0.5)

. Then

τ = - 0.5

Δ = 1

. Discriminant

τ^{2} - 4Δ = 0.25 - 4 = - 3.75 < 0

, and

τ < 0

⇒ stable spiral. The pendulum oscillates with decaying amplitude back to rest. ✔ (physically correct: friction damps swings.) Step 4 — At $(π, 0)$ :

cos π = - 1

, so

A = (01 1 - 0.5)

. Then

τ = - 0.5

Δ = (0) (- 0.5) - (1) (1) = - 1 < 0

⇒ saddle. The inverted equilibrium is unstable; only one special trajectory (stable manifold) approaches it.

(0, 0) : stable spiral (π, 0) : saddle

The full phase portrait: spirals winding into each "down" position, saddles at each "up" position, with separatrices dividing rocking from full-rotation motions.

Common Misconceptions

A center in the linearization need NOT be a center in the nonlinear system. When

τ = 0

(purely imaginary eigenvalues), the fixed point is non-hyperbolic; Hartman–Grobman does not apply. Higher-order nonlinear terms can turn a linear "center" into a slowly winding stable or unstable spiral. Always check borderline cases with extra tools (Lyapunov functions, normal forms). Fixed points are not always isolated, and not all systems have any. If

det A = 0

the linearization fails to give an isolated equilibrium (lines of fixed points possible). And

\overset{x}{˙} = 1

has no fixed point at all — the flow simply translates.

Nonlinear destroys a linear center

Consider

\overset{x}{˙} = - y + a x (x^{2} + y^{2}), \overset{y}{˙} = x + a y (x^{2} + y^{2}) .

The Jacobian at the origin is

(01 - 1 0)

with

τ = 0, Δ = 1

— the linearization predicts a center. But converting to polar coordinates (

r^{2} = x^{2} + y^{2}

) gives

\overset{r}{˙} = a r^{3}

. For

a > 0

the radius grows (unstable spiral); for

a < 0

it shrinks (stable spiral); only

a = 0

is a true center. The linear prediction is simply wrong about stability — exactly the failure mode the Hartman–Grobman Theorem warns about.

Connections

Phase portraits sit at the crossroads of several theories:

Eigenvalues and Eigenvectors supply the entire local classification through $τ, Δ$ and the eigendirections.
Stability Theory and Lyapunov Functions handle the non-hyperbolic cases the linearization cannot, and certify global stability.
Bifurcation Theory studies how the number and type of fixed points change as parameters vary — fixed points are born, collide, and change stability (saddle-node, Hopf, pitchfork bifurcations).
Limit Cycles and the Poincaré–Bendixson Theorem organize bounded planar trajectories that don't reach an equilibrium — the only options in 2D are equilibria, closed orbits, or homoclinic/heteroclinic connections.
In higher dimensions, fixed points connect to Stable and Unstable Manifolds and ultimately to Chaos and Strange Attractors.

Real-World Applications

Ecology — Lotka–Volterra predator–prey: the coexistence equilibrium is a center (in the classic model), giving the observed population oscillations of lynx and hare.
Neuroscience — FitzHugh–Nagumo / Hodgkin–Huxley reductions: neuron firing is a limit cycle around an unstable spiral; threshold behaviour is governed by a saddle's stable manifold.
Mechanical engineering — damped oscillators and pendulums: stable spirals represent settling structures; saddles mark unstable inverted configurations (Segway/rocket balancing).
Electrical engineering — Van der Pol oscillator: an unstable spiral at the origin plus a stable limit cycle models vacuum-tube and electronic relaxation oscillators.
Epidemiology — SIR models: disease-free and endemic equilibria; the basic reproduction number $R_{0}$ determines which is stable, a transcritical bifurcation.
Chemistry — Belousov–Zhabotinsky / Brusselator reactions: oscillating concentrations arise from a Hopf bifurcation turning a stable spiral into a limit cycle.
Control systems: controller design literally reshapes the Jacobian's eigenvalues to push fixed points into the stable (negative-trace) region.

Historical Development

1673 — Huygens analyzes the pendulum and isochronism, an early study of phase-space-like oscillation, though without the geometric framework.
1834–1836 — Hamilton recasts mechanics in $(q, p)$ phase space, the conceptual ancestor of the modern state space.
1881–1886 — Poincaré publishes Sur les courbes définies par une équation différentielle, founding qualitative theory: classification of singular points, indices, and the return map.
1892 — Lyapunov's thesis The General Problem of the Stability of Motion introduces Lyapunov functions and rigorous stability definitions, handling cases the linearization misses.
1899 — Poincaré completes Les méthodes nouvelles de la mécanique céleste, discovering homoclinic tangles (the seed of chaos).
1901 — Ivar Bendixson proves the negative criterion and (with Poincaré) the Poincaré–Bendixson theorem on planar limit cycles.
1920s–1937 — Andronov and the Gorky school connect phase portraits to engineering oscillations, formalize structural stability with Pontryagin.
1959–1960 — Hartman and Grobman independently prove the local linearization (conjugacy) theorem for hyperbolic fixed points.
1963 — Lorenz numerically discovers a strange attractor, dramatizing how fixed-point analysis generalizes to chaotic flows.

Subtleties & Limits

Non-hyperbolic and global questions remain hard.

At non-hyperbolic fixed points (zero-real-part eigenvalues), no general algorithm classifies behaviour; center manifold theory and normal-form reduction help but can require infinitely many terms to decide stability.
In $n \geq 3$ dimensions, the simple trace–determinant table fails; eigenvalue configurations multiply, and global structure can be chaotic. Determining the exact basin of attraction of a fixed point is generally undecidable in closed form.
The number of limit cycles of planar polynomial systems is Hilbert's 16th problem — still open even for quadratic vector fields.
Reconstructing reliable phase portraits from noisy experimental data (delay embedding, attractor reconstruction) is an active applied frontier.