The Central Idea
A phase portrait is a geometric summary of a dynamical system in state space.
It combines:
- equilibria;
- direction information;
- representative trajectories;
- stable and unstable structures;
- long-term patterns.
Phase Space
The term phase space is often used interchangeably with state space in introductory dynamical systems.
For a two-variable system,
\[ \mathbf{x}(t)= \begin{pmatrix} x(t)\\ y(t) \end{pmatrix}, \]the phase plane is the \(xy\)-plane.
Every point
\[ (x,y) \]represents one possible state.
What a Phase Portrait Shows
A phase portrait does not usually show time on one of its axes.
Instead, the axes represent state variables, and time is indicated by the direction in which trajectories move.
Arrows along trajectories show the direction of increasing time.
Continuous Two-Dimensional Systems
A standard autonomous system has the form
\[ \frac{dx}{dt}=f(x,y), \] \[ \frac{dy}{dt}=g(x,y). \]In vector form,
\[ \boxed{ \frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}) }. \]The vector field \(\mathbf{F}\) assigns a direction and speed to every state.
The Vector Field
At a state \((x,y)\), the vector
\[ \left( \frac{dx}{dt}, \frac{dy}{dt} \right) = \bigl(f(x,y),g(x,y)\bigr) \]tells us the instantaneous state-space velocity.
A phase portrait is built by following these directions through the state space.
Example: Reading a Vector
Suppose
\[ \frac{dx}{dt}=x-y, \] \[ \frac{dy}{dt}=x+y. \]At
\[ (x,y)=(2,1), \]we have
\[ \frac{dx}{dt}=2-1=1, \] \[ \frac{dy}{dt}=2+1=3. \]The local vector is
\[ \boxed{(1,3)}. \]The state moves toward increasing \(x\) and increasing \(y\).
Direction Fields in the Phase Plane
A two-dimensional vector field can be visualized by placing small arrows at selected points.
Each arrow indicates:
- the direction of motion;
- the relative speed of motion;
- how nearby trajectories are likely to bend.
The collection of arrows provides the local structure from which trajectories emerge.
Trajectories Follow the Vector Field
A trajectory
\[ \mathbf{x}(t) \]must be tangent to the vector field at every point it passes through.
In symbols,
\[ \frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}(t)). \]The tangent vector to the trajectory is exactly the vector prescribed by the dynamical system.
Equilibria in a Phase Portrait
An equilibrium satisfies
\[ f(x^\ast,y^\ast)=0 \]and
\[ g(x^\ast,y^\ast)=0. \]Therefore,
\[ \mathbf{F}(x^\ast,y^\ast)=\mathbf{0}. \]At an equilibrium, the vector field has zero magnitude.
Equilibria Organize the Portrait
Nearby trajectories often curve toward, away from, around, or past an equilibrium.
The local geometry around each equilibrium provides a first classification of the phase portrait.
Common equilibrium types include:
- stable nodes;
- unstable nodes;
- saddles;
- stable spirals;
- unstable spirals;
- centers.
Stable Node
A stable node is an equilibrium toward which nearby trajectories move.
A simple example is
\[ x'=-x, \qquad y'=-2y. \]The solutions are
\[ x(t)=x_0e^{-t}, \qquad y(t)=y_0e^{-2t}. \]Both coordinates approach zero, so trajectories move toward the origin.
Geometry of a Stable Node
In
\[ x'=-x, \qquad y'=-2y, \]every trajectory approaches
\[ (0,0). \]The \(y\)-component decays faster than the \(x\)-component because
\[ e^{-2t} \]decreases faster than
\[ e^{-t}. \]This affects the shape and orientation of trajectories near the origin.
Unstable Node
Consider
\[ x'=x, \qquad y'=2y. \]Then
\[ x(t)=x_0e^t, \qquad y(t)=y_0e^{2t}. \]Except for the equilibrium itself, nearby trajectories move away from the origin.
The origin is an unstable node.
Saddle Point
Consider
\[ x'=-x, \qquad y'=y. \]Then
\[ x(t)=x_0e^{-t}, \qquad y(t)=y_0e^t. \]Motion contracts along the \(x\)-direction and expands along the \(y\)-direction.
The origin is a saddle.
Stable and Unstable Directions of a Saddle
For
\[ x'=-x, \qquad y'=y, \]points on the \(x\)-axis satisfy
\[ y=0, \]and move toward the origin.
Points on the \(y\)-axis satisfy
\[ x=0, \]and move away from the origin in forward time.
These directions form the simplest examples of stable and unstable manifolds.
Stable Spiral
A stable spiral occurs when trajectories rotate while moving inward toward an equilibrium.
A simple example is
\[ x'=-x-y, \] \[ y'=x-y. \]The system combines rotation with exponential decay.
Trajectories spiral toward the origin.
Unstable Spiral
Consider
\[ x'=x-y, \] \[ y'=x+y. \]The rotational component remains, but the amplitude grows.
Trajectories spiral outward from the origin.
The origin is an unstable spiral.
Center
Consider the harmonic system
\[ x'=y, \qquad y'=-x. \]The quantity
\[ x^2+y^2 \]is conserved.
Therefore trajectories lie on circles around the origin.
The origin is called a center.
Centers versus Spirals
Both centers and spirals involve rotation around an equilibrium.
The difference is radial behavior.
- A center has nearby closed trajectories with no inward or outward drift.
- A stable spiral rotates inward.
- An unstable spiral rotates outward.
This distinction is often visible immediately in a phase portrait.
Eigenvalues Predict Linear Phase Portraits
For the linear system
\[ \mathbf{x}'=A\mathbf{x}, \]the eigenvalues of \(A\) strongly determine the local geometry.
Their signs, real parts, and imaginary parts tell us whether trajectories contract, expand, rotate, or combine these behaviors.
Real Negative Eigenvalues
If a two-dimensional linear system has two real negative eigenvalues, the origin is typically a stable node.
For example,
\[ \lambda_1=-1, \qquad \lambda_2=-4. \]Both eigendirections contract in forward time.
Real Positive Eigenvalues
If both eigenvalues are real and positive, trajectories expand in both eigendirections.
The origin is typically an unstable node.
For example,
\[ \lambda_1=2, \qquad \lambda_2=5. \]Opposite-Sign Eigenvalues
If one eigenvalue is negative and the other positive,
\[ \lambda_1<0<\lambda_2, \]the equilibrium is a saddle.
One eigendirection is stable and the other is unstable.
Complex Eigenvalues
Suppose
\[ \lambda=\alpha\pm i\beta \]with
\[ \beta\ne0. \]The imaginary part produces rotational behavior.
The real part determines radial growth or decay.
Complex Eigenvalue Cases
- If \(\alpha<0\), trajectories spiral inward.
- If \(\alpha>0\), trajectories spiral outward.
- If \(\alpha=0\) in an ideal linear system, closed orbits may occur.
Thus
\[ \boxed{ \text{real part controls growth or decay} } \]while
\[ \boxed{ \text{imaginary part produces rotation} }. \]Nullclines
For
\[ x'=f(x,y), \qquad y'=g(x,y), \]the \(x\)-nullcline satisfies
\[ f(x,y)=0. \]Along this curve, the horizontal component of motion is zero.
The \(y\)-nullcline satisfies
\[ g(x,y)=0, \]so the vertical component is zero.
Why Nullclines Are Useful
Nullclines divide the phase plane into regions where the signs of
\[ x' \]and
\[ y' \]are constant.
This makes it possible to infer the general direction of the vector field without solving the differential equations exactly.
Example: Nullclines
Consider
\[ x'=x(2-y), \] \[ y'=y(x-1). \]The \(x\)-nullclines are
\[ x=0 \qquad\text{and}\qquad y=2. \]The \(y\)-nullclines are
\[ y=0 \qquad\text{and}\qquad x=1. \]Equilibria from Nullclines
Equilibria occur where an \(x\)-nullcline and a \(y\)-nullcline intersect.
In the example,
\[ x'=x(2-y), \qquad y'=y(x-1), \]the equilibria are
\[ \boxed{(0,0)} \]and
\[ \boxed{(1,2)}. \]Sign Charts in the Phase Plane
After drawing the nullclines, choose a test point in each region.
Determine the signs of
\[ x' \qquad\text{and}\qquad y'. \]For example:
- \((+,+)\) means motion right and up;
- \((+,-)\) means motion right and down;
- \((-,+)\) means motion left and up;
- \((-,-)\) means motion left and down.
Trajectories Usually Do Not Cross
For ordinary differential equations satisfying standard uniqueness conditions, distinct trajectories cannot cross at the same state.
If two trajectories crossed, the same state would have two different future paths.
That would contradict uniqueness.
This restriction helps phase portraits maintain a coherent geometry.
Invariant Curves
An invariant curve is a curve that trajectories remain on once they start there.
Eigenvector lines in linear systems are important examples.
Conserved level sets can also form invariant curves.
Separatrices
A separatrix is a special trajectory or invariant curve that separates qualitatively different kinds of motion.
Separatrices often arise from saddle equilibria.
They may divide the state space into different basins of attraction or different dynamical regions.
Example: Basin Boundary
Suppose a system has two attracting equilibria.
Some initial states move toward the first, while others move toward the second.
A separatrix can form the boundary between these two basins of attraction.
Small changes near that boundary can therefore change the long-term outcome.
Periodic Orbits in Phase Portraits
A periodic orbit appears as a closed trajectory in the phase plane.
If
\[ \mathbf{x}(t+T)=\mathbf{x}(t), \]then the system returns to the same state after period \(T\).
A closed orbit can represent recurring oscillation in the original model.
Closed Curves Are Not Always Equilibria
An equilibrium is one stationary point.
A periodic orbit is a moving trajectory that repeatedly traces a closed curve.
These are fundamentally different dynamical structures.
Limit Cycles
A limit cycle is an isolated periodic orbit.
Nearby trajectories may approach it or move away from it.
A stable limit cycle represents a self-sustaining oscillation toward which nearby motions converge.
Limit cycles will connect later to periodic-orbit analysis.
Phase Portraits and Long-Term Behavior
A phase portrait allows us to ask where trajectories go as time becomes large.
They may:
- approach an equilibrium;
- approach a periodic orbit;
- remain on a conserved curve;
- escape to infinity;
- approach a more complicated attractor.
Phase Portraits without Exact Solutions
One of the major advantages of phase-plane analysis is that exact formulas for
\[ x(t) \qquad\text{and}\qquad y(t) \]are not always necessary.
Equilibria, nullclines, sign information, local linearization, conserved quantities, and numerical trajectories can often reveal the essential geometry.
Numerical Phase Portraits
Computers can approximate many trajectories from different initial conditions.
For
\[ \mathbf{x}'=\mathbf{F}(\mathbf{x}), \]a numerical solver produces approximate states
\[ \mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \ldots \]which can be plotted in the phase plane.
Repeating this process for many initial states builds a numerical phase portrait.
Phase Portrait versus Time-Series Plot
| Feature | Time-Series Plot | Phase Portrait |
|---|---|---|
| Axes | Time and one state variable | State variable against state variable |
| Focus | Change through time | Geometry of state evolution |
| Equilibria | Constant levels | Stationary points |
| Periodic behavior | Repeating waveform | Closed trajectory |
| Basins | Usually difficult to see | Often geometrically visible |
Example: Damped Oscillator
Consider
\[ x'=v, \] \[ v'=-x-cv, \]with
\[ c>0. \]The state is
\[ (x,v). \]Depending on the damping level, trajectories may spiral or curve inward toward the equilibrium
\[ (0,0). \]Physical Interpretation of a Spiral
In a damped mechanical system, each loop in the phase plane corresponds to another oscillation.
As the trajectory spirals inward, the amplitude decreases.
Thus the geometry of the phase portrait directly expresses energy loss and decay of oscillation.
Example: Predator-Prey Geometry
A predator-prey model might use the state
\[ (R,F), \]where \(R\) is prey population and \(F\) is predator population.
Closed or nearly closed phase-plane trajectories can represent cycles in which one population rises, the other responds, and the pattern repeats.
The portrait shows the interaction of both populations simultaneously.
Example: Competing Stable States
Some nonlinear systems contain multiple attracting equilibria.
A phase portrait may show trajectories in one region moving toward one equilibrium and trajectories elsewhere moving toward another.
The boundary between those regions becomes dynamically important because it separates different long-term outcomes.
Local and Global Information
A phase portrait can contain both local and global information.
Local information concerns behavior near an equilibrium.
Global information concerns how trajectories move across larger portions of the state space.
A locally stable equilibrium may still have only a limited basin of attraction.
Linear Phase Portraits
Linear systems
\[ \mathbf{x}'=A\mathbf{x} \]are especially useful because their phase portraits can often be classified directly from eigenvalues and eigenvectors.
These linear portraits serve as templates for understanding nonlinear systems near equilibria.
Nonlinear Phase Portraits
Nonlinear systems may contain features impossible in simple linear systems, including:
- multiple isolated equilibria;
- curved nullclines;
- multiple basins of attraction;
- limit cycles;
- homoclinic or heteroclinic connections;
- complex attractors.
Their phase portraits can therefore be much richer.
Local Linear Approximation
Even when a system is nonlinear,
\[ \mathbf{x}'=\mathbf{F}(\mathbf{x}), \]its behavior near an equilibrium can often be approximated by a linear system.
This approximation is built from the derivative matrix, or Jacobian, of \(\mathbf{F}\).
The next lesson develops this idea systematically.
Common Error: Drawing Trajectories That Cross
In systems with unique solutions, distinct trajectories should not cross in the phase plane.
A hand-drawn portrait that contains arbitrary trajectory crossings is usually inconsistent with the underlying differential equation.
Common Error: Forgetting Direction Arrows
A geometric curve alone does not tell us which way the system moves.
The same curve followed in opposite directions represents different dynamics.
Phase portraits should therefore include direction arrows or another clear indication of increasing time.
Common Error: Confusing Nullclines with Trajectories
A nullcline is a curve on which one component of the vector field vanishes.
It is not automatically a trajectory.
A trajectory remains on a nullcline only if the vector field is tangent to that curve in a way that preserves it.
Common Error: Assuming Every Closed Curve Is a Limit Cycle
A closed trajectory is periodic.
A limit cycle is an isolated periodic orbit.
The harmonic oscillator has a continuous family of closed circles, so those circles are periodic orbits but not isolated limit cycles.
Common Error: Classifying an Equilibrium from Appearance Alone
A numerical plot can be suggestive, but classification should be supported by the equations.
Eigenvalues, sign analysis, nullclines, conserved quantities, or other mathematical evidence should confirm the portrait whenever possible.
Common Error: Ignoring the State-Space Domain
A phase portrait should respect the admissible state space.
If \(x\) and \(y\) represent populations, negative values may be mathematically drawable but physically irrelevant.
Model constraints should guide which region of the portrait is interpreted.
A Reliable Procedure for Sketching a Phase Portrait
- Write the system in first-order form.
- Identify the relevant state-space domain.
- Find all equilibria.
- Find the nullclines.
- Determine the direction of the vector field in representative regions.
- Analyze the local behavior near equilibria.
- Identify invariant lines or curves when available.
- Sketch representative trajectories.
- Add arrows showing increasing time.
- Identify periodic orbits, separatrices, basins, and long-term behavior.
A Reliable Procedure for Reading a Phase Portrait
- Identify the state variables on the axes.
- Locate every equilibrium.
- Determine which equilibria attract or repel nearby trajectories.
- Look for saddle behavior and special invariant directions.
- Identify closed trajectories or spirals.
- Trace the direction of increasing time.
- Look for separatrices or basin boundaries.
- Determine the likely long-term outcome from different initial states.
- Translate the geometric conclusions back into the original application.
Practice 1: Vector at a Point
Consider
\[ x'=x+y, \qquad y'=x-y. \]Find the vector field value at
\[ (x,y)=(2,1). \]Practice 2: Equilibrium
Find the equilibrium of
\[ x'=3-x, \qquad y'=4-2y. \]Practice 3: Stable Node
Classify the origin for
\[ x'=-2x, \qquad y'=-5y. \]Practice 4: Saddle
Classify the origin for
\[ x'=-x, \qquad y'=3y. \]Practice 5: Center
Consider
\[ x'=y, \qquad y'=-x. \]Explain why trajectories satisfy
\[ x^2+y^2=C. \]Practice 6: Nullclines
Find the nullclines of
\[ x'=x(4-y), \] \[ y'=y(x-2). \]Practice 7: Equilibria from Nullclines
Using the previous system,
\[ x'=x(4-y), \qquad y'=y(x-2), \]find all equilibria.
Practice 8: Spiral Classification
A linear system has eigenvalues
\[ -2\pm3i. \]What qualitative equilibrium type should you expect?
Practice 9: Unstable Spiral
A linear system has eigenvalues
\[ 1\pm4i. \]What qualitative behavior should nearby trajectories show?
Practice 10: Identify the Error
A student draws two distinct solution trajectories crossing at the point
\[ (1,2) \]for a smooth autonomous differential equation with unique solutions.
Explain why the drawing is inconsistent.
Practice Answers
Practice 1
At
\[ (2,1), \]we have
\[ x'=2+1=3, \] \[ y'=2-1=1. \]Therefore,
\[ \boxed{(x',y')=(3,1)}. \]Practice 2
Set both derivatives equal to zero:
\[ 3-x=0, \] \[ 4-2y=0. \]Thus
\[ \boxed{(x^\ast,y^\ast)=(3,2)}. \]Practice 3
Both coordinates decay exponentially:
\[ x(t)=x_0e^{-2t}, \qquad y(t)=y_0e^{-5t}. \]Therefore the origin is a
\[ \boxed{\text{stable node}}. \]Practice 4
The \(x\)-direction contracts because
\[ x(t)=x_0e^{-t}, \]while the \(y\)-direction expands because
\[ y(t)=y_0e^{3t}. \]Therefore the origin is a
\[ \boxed{\text{saddle}}. \]Practice 5
Differentiate
\[ x^2+y^2. \]Then
\[ \frac{d}{dt}(x^2+y^2) = 2xx'+2yy'. \]Substitute
\[ x'=y, \qquad y'=-x: \] \[ 2xy+2y(-x)=0. \]Therefore \(x^2+y^2\) is constant along trajectories.
Practice 6
The \(x\)-nullclines satisfy
\[ x(4-y)=0, \]so
\[ \boxed{x=0\quad\text{or}\quad y=4}. \]The \(y\)-nullclines satisfy
\[ y(x-2)=0, \]so
\[ \boxed{y=0\quad\text{or}\quad x=2}. \]Practice 7
The simultaneous nullcline intersections are
\[ \boxed{(0,0)} \]and
\[ \boxed{(2,4)}. \]Practice 8
The eigenvalues have negative real part and nonzero imaginary part.
Therefore trajectories rotate while decaying toward the equilibrium.
\[ \boxed{\text{stable spiral}}. \]Practice 9
The eigenvalues have positive real part and nonzero imaginary part.
Therefore trajectories rotate while moving away from the equilibrium.
\[ \boxed{\text{unstable spiral}}. \]Practice 10
If two distinct trajectories crossed at the same state, then the same state would have two different future solution paths.
That contradicts uniqueness for the assumed differential equation.
Therefore the two distinct trajectories should not cross.
What to Remember
A phase portrait shows the geometry of a dynamical system in state space.
\[ \boxed{ \text{phase portrait} = \text{equilibria} + \text{direction field} + \text{representative trajectories} } \]- equilibria appear as stationary points;
- stable nodes attract nearby trajectories;
- unstable nodes repel them;
- saddles attract in some directions and repel in others;
- spirals combine rotation with inward or outward motion;
- centers are surrounded by persistent closed motion;
- nullclines help divide the plane into directional regions;
- separatrices can divide different basins of attraction;
- periodic orbits appear as closed trajectories;
- phase portraits often reveal long-term behavior without exact solutions.
The essential question is not merely where one trajectory goes, but how the entire state space is organized by the dynamical law.
Where We Go Next
For linear systems, eigenvalues and eigenvectors often determine the local phase portrait directly.
Nonlinear systems are more complicated, but near an equilibrium they can often be approximated by a linear system built from derivatives.
The next lesson, Linearization near Equilibria, develops the Jacobian matrix and shows how local nonlinear dynamics can often be understood through a simpler linear approximation.