Phase Portraits, Equilibria, and Local Stability

How autonomous systems are read geometrically in the phase plane, why equilibria organize nearby motion, and how linearization gives the first local stability test.
Modified

April 26, 2026

Keywords

phase portrait, equilibrium, local stability, linearization, Jacobian

1 Role

This is the fourth page of the ODEs and Dynamical Systems module.

Its job is to move from exact linear systems into the geometric reading of nonlinear autonomous systems:

draw the flow, find the equilibria, and use linearization to understand what nearby trajectories do

2 First-Pass Promise

Read this page after Linear Systems, Matrix Exponentials, and Modes.

If you stop here, you should still understand:

  • what a phase portrait represents
  • why equilibria are the first points to inspect
  • how the Jacobian linearization gives a first local stability test
  • what sinks, sources, saddles, and centers mean geometrically

3 Why It Matters

Closed-form solutions are rare once systems become nonlinear.

But we can still learn a lot from geometry.

For an autonomous system

\[ y' = f(y), \]

the key questions are:

  • where are the equilibria
  • which directions does the vector field point
  • which trajectories move toward or away from an equilibrium
  • when does local behavior look approximately linear

This is why phase portraits matter. They turn differential equations into pictures of possible motion.

That viewpoint is essential in:

  • mechanics and oscillators
  • predator-prey and population models
  • control and feedback systems
  • local dynamics near operating points
  • continuous-time ML and flow-based models

4 Prerequisite Recall

  • the previous page explained how linear systems are organized by modes and eigenvalues
  • an autonomous system has no explicit time dependence in the right-hand side
  • the Jacobian matrix is the best local linear map for a vector-valued function
  • uniqueness prevents two distinct trajectories from crossing through the same state

5 Intuition

5.1 The Phase Plane Tracks State, Not Time On The Horizontal Axis

For a two-dimensional autonomous system

\[ \begin{aligned} x' &= f(x,y),\\ y' &= g(x,y), \end{aligned} \]

the phase plane is the (x,y) state plane.

Each point is a state, and the vector field tells us which way the trajectory moves from there.

5.2 Equilibria Are The First Landmarks

An equilibrium is a state where the vector field vanishes.

If a trajectory starts there, it stays there.

Even more importantly, nearby trajectories are often organized by what the system does around those points.

5.3 Linearization Is The First Local Approximation

Near an equilibrium y_*, the nonlinear system can often be approximated by

\[ u' = J_f(y_*)u, \]

where J_f(y_*) is the Jacobian matrix.

That means the linear systems page becomes a local tool for nonlinear systems.

5.4 Geometric Types Come From Eigenvalues

For a planar linearized system:

  • both eigenvalues with negative real parts suggest a local sink
  • both with positive real parts suggest a local source
  • mixed signs suggest a saddle
  • purely imaginary eigenvalues are a borderline case where linearization alone is not always decisive

This is why local stability is really a blend of geometry and linear algebra.

6 Formal Core

Definition 1 (Definition: Phase Portrait) A phase portrait is the collection of trajectories, or at least a qualitative sketch of them, in state space for an autonomous system.

It shows how states move, not how coordinates vary directly with time on a graph.

Definition 2 (Definition: Equilibrium) For an autonomous system

\[ y' = f(y), \]

an equilibrium y_* satisfies

\[ f(y_*)=0. \]

Definition 3 (Definition: Local Stability) An equilibrium is locally stable if trajectories that start sufficiently close stay close for future time, and asymptotically stable if they also converge to the equilibrium.

Theorem 1 (Theorem Idea: Linearization Near An Equilibrium) If f is differentiable and y_* is an equilibrium, then near y_* the system

\[ y' = f(y) \]

is approximated by the linear system

\[ u' = J_f(y_*)u, \qquad u = y-y_*. \]

At first pass, the main lesson is:

  • the Jacobian is the local linear dynamical rule
  • the eigenvalues of that Jacobian give the first local classification

Theorem 2 (Theorem Idea: Hyperbolic Equilibria Are Classified By The Linearization) If the Jacobian at an equilibrium has no eigenvalues with zero real part, then the linearized system gives the correct first local type: sink, source, or saddle.

This is the safe first-pass regime.

Borderline cases such as centers or zero-real-part eigenvalues need extra care.

7 A Small Worked Example

Consider the nonlinear system

\[ \begin{aligned} x' &= y,\\ y' &= -x - 0.3y + x^2. \end{aligned} \]

7.1 Step 1: Find The Equilibria

At equilibrium we need

\[ y=0, \qquad -x + x^2 = 0. \]

So

\[ x(x-1)=0, \]

and the equilibria are

\[ (0,0)\quad\text{and}\quad (1,0). \]

7.2 Step 2: Compute The Jacobian

The Jacobian is

\[ J_f(x,y)= \begin{bmatrix} 0 & 1\\ -1+2x & -0.3 \end{bmatrix}. \]

At (0,0) we get

\[ J_f(0,0)= \begin{bmatrix} 0 & 1\\ -1 & -0.3 \end{bmatrix}. \]

This has eigenvalues with negative real part, so the origin is locally attracting: a spiral sink.

At (1,0) we get

\[ J_f(1,0)= \begin{bmatrix} 0 & 1\\ 1 & -0.3 \end{bmatrix}. \]

This has eigenvalues of opposite sign, so (1,0) is a saddle.

7.3 Step 3: Read The Local Phase Portrait

Without solving the nonlinear system explicitly, we already know:

  • trajectories near (0,0) spiral inward
  • trajectories near (1,0) are repelled in one direction and attracted in another
  • the phase portrait must reflect both a locally attracting equilibrium and a saddle geometry

That is exactly the kind of information phase portraits are built to show.

8 Computation Lens

A phase portrait is the exact qualitative object. Numerical solvers only approximate trajectories that live inside it.

So when we simulate a nonlinear system:

  • equilibria tell us what structures should appear
  • linearization predicts local behavior near those points
  • time stepping must be interpreted against that qualitative backdrop

This is why Time-Stepping for ODEs and Stability and phase portraits complement each other instead of competing.

9 Application Lens

9.1 Mechanics And Oscillation

Damped oscillators, pendula, and coupled mechanical systems are naturally read in the phase plane.

9.2 Control And Operating Points

Control theory often studies systems near an equilibrium or operating point, where linearization is the first approximation.

9.3 Continuous-Time ML

Flow-based models, neural ODEs, and optimization dynamics are often interpreted through fixed points, local attraction, and stability language.

10 Stop Here For First Pass

If you can now explain:

  • what a phase portrait is
  • how to find equilibria
  • why the Jacobian linearization is the first local test
  • what sink, source, and saddle mean geometrically

then this page has done its job.

11 Go Deeper

After this page, the next natural step is:

The strongest adjacent pages are:

12 Optional Deeper Reading After First Pass

The strongest current references connected to this page are:

13 Sources and Further Reading

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