Introduction

In the previous lesson, we learned that systems of differential equations can be written as:

$$\frac{dX}{dt}=AX$$

We also learned that eigenvalues determine whether solutions:

• Grow
• Decay
• Oscillate

However, equations alone often fail to provide intuition.

When engineers, statisticians, epidemiologists, and operations researchers study dynamical systems, they often ask:

What does the system actually do?

This is where phase planes become useful.

A phase plane allows us to visualize the entire behavior of a system without solving the differential equations.

In this lesson we will learn:

• What a phase plane is
• What trajectories are
• What vector fields are
• Stable nodes
• Unstable nodes
• Saddle points
• Spiral points

These ideas will eventually lead us toward stochastic differential equations, where randomness is added to the dynamics.

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Why We Need Phase Planes

Consider the system:

$$\frac{dx}{dt}=2x+y$$

$$\frac{dy}{dt}=x+2y$$

Looking at these equations directly does not immediately tell us:

• Where the system is going
• Whether it stabilizes
• Whether it oscillates

A phase plane gives us a picture of the dynamics.

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What Is a Phase Plane?

Suppose our system contains two variables:

$$x$$

and

$$y$$

We draw:

• x on the horizontal axis
• y on the vertical axis

Every point on this graph represents one possible state of the system.

For example:

$$(x,y)=(5,3)$$

means:

• x equals 5
• y equals 3

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Supply Chain Example

Suppose:

$$x=Inventory$$

and

$$y=Sales$$

Then every point in the phase plane represents a possible combination of:

• Inventory
• Sales

For example:

$$(1000,200)$$

might represent:

• 1000 units of inventory
• 200 units of monthly sales

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Health Example

Suppose:

$$x=Healthy\ Cells$$

and

$$y=Infected\ Cells$$

Then each point represents a possible health state.

For example:

$$(9000,1000)$$

means:

• 9000 healthy cells
• 1000 infected cells

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Business Example

Suppose:

$$x=Advertising$$

and

$$y=Customers$$

Every point represents a possible business state.

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What Is a Trajectory?

Suppose the system starts at:

$$(1,2)$$

As time passes:

• x changes
• y changes

The point moves through the phase plane.

The path traced by the moving point is called a trajectory.

A trajectory is simply:

The history of the system through time.

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Example

Suppose inventory and sales evolve as:

$$(1000,200)$$

then

$$(950,220)$$

then

$$(900,250)$$

then

$$(870,280)$$

Plotting these points creates a trajectory.

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Vector Fields

A trajectory shows one possible path.

A vector field shows all possible paths.

At every point in the phase plane:

$$(x,y)$$

the differential equations determine:

$$\frac{dx}{dt}$$

and

$$\frac{dy}{dt}$$

These derivatives form a small arrow.

The arrow tells us:

• direction of motion
• speed of motion

Plotting arrows everywhere creates a vector field.

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Why Vector Fields Matter

A vector field is like a weather map.

Instead of showing wind direction:

it shows system direction.

Without solving anything, we can see how the system behaves.

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Equilibrium Points

Recall from Lesson 4:

An equilibrium occurs when all derivatives equal zero.

For a system:

$$\frac{dx}{dt}=0$$

and

$$\frac{dy}{dt}=0$$

must both hold.

At equilibrium:

Nothing changes.

The system remains there forever.

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Example

Consider:

$$\frac{dx}{dt}=-x$$

$$\frac{dy}{dt}=-y$$

Setting both derivatives to zero gives:

$$x=0$$

$$y=0$$

Therefore the equilibrium is:

$$(0,0)$$

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Stable Nodes

Suppose:

$$\frac{dx}{dt}=-x$$

$$\frac{dy}{dt}=-2y$$

The solutions are:

$$x(t)=C_1e^{-t}$$

$$y(t)=C_2e^{-2t}$$

Both variables approach zero.

All trajectories move toward:

$$(0,0)$$

This equilibrium is called a stable node.

---

Supply Chain Interpretation

Inventory deviations disappear.

Sales deviations disappear.

The business settles into equilibrium.

---

Health Interpretation

Infection levels decline.

The patient recovers.

---

Business Interpretation

Advertising and customer acquisition stabilize.

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Unstable Nodes

Now consider:

$$\frac{dx}{dt}=x$$

$$\frac{dy}{dt}=2y$$

Solutions become:

$$x(t)=C_1e^t$$

$$y(t)=C_2e^{2t}$$

Everything grows.

Trajectories move away from equilibrium.

This is an unstable node.

---

Supply Chain Interpretation

Inventory grows uncontrollably.

---

Health Interpretation

Disease spreads rapidly.

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Business Interpretation

Growth accelerates without bound.

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Saddle Points

Consider:

$$\frac{dx}{dt}=x$$

$$\frac{dy}{dt}=-y$$

One direction grows.

One direction shrinks.

Some trajectories move toward equilibrium.

Others move away.

This creates a saddle point.

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Why Called a Saddle?

Imagine sitting on a horse saddle.

One direction curves upward.

Another curves downward.

The equilibrium behaves similarly.

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Supply Chain Interpretation

Certain inventory-sales patterns stabilize.

Others become unstable.

---

Health Interpretation

Some infections clear naturally.

Others worsen.

Small differences in conditions lead to very different outcomes.

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Spiral Points

Consider:

$$\frac{dx}{dt}=-y$$

$$\frac{dy}{dt}=x$$

The trajectory circles around the origin.

The system oscillates forever.

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Engineering Interpretation

A frictionless spring oscillates forever.

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Health Interpretation

Certain biological rhythms oscillate naturally.

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Supply Chain Interpretation

Inventory repeatedly overshoots and undershoots target levels.

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Stable Spirals

Consider:

$$\frac{dx}{dt}=-x-y$$

$$\frac{dy}{dt}=x-y$$

Now oscillation still occurs.

However, the amplitude gradually shrinks.

Trajectories spiral inward.

Eventually they reach equilibrium.

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Supply Chain Interpretation

Inventory corrections become smaller and smaller until stability is reached.

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Health Interpretation

Inflammatory markers fluctuate before normalizing.

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Business Interpretation

Sales cycles dampen and eventually stabilize.

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Attractors

An attractor is a state toward which nearby trajectories move.

Stable equilibria are examples of attractors.

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Supply Chain Example

Desired inventory level.

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Health Example

Recovery.

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Business Example

Long-term market equilibrium.

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Why Statisticians Care About Phase Planes

Phase plane ideas appear throughout statistics.

Examples include:

State-Space Models

The hidden state moves through a phase space.

Kalman Filters

State estimates evolve through time.

Epidemiology

Disease compartments form trajectories.

Bayesian Dynamic Models

Latent states evolve through phase space.

Stochastic Differential Equations

Random trajectories move through a stochastic phase plane.

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Connection to Eigenvalues

The behavior we observe in phase planes is determined by eigenvalues.

Negative Eigenvalues

Stable node.

Positive Eigenvalues

Unstable node.

Mixed Signs

Saddle point.

Complex Eigenvalues

Spirals and oscillations.

This is why eigenvalues are so important.

They determine the geometry of the system.

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The Big Picture

A phase plane transforms equations into pictures.

Instead of solving differential equations, we can visualize:

• Stability
• Oscillation
• Growth
• Decay
• Long-term behavior

This geometric viewpoint is one of the most powerful ideas in dynamical systems.

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Looking Ahead

In Lesson 9 we will study:

Nonlinear Dynamical Systems

We will learn why real-world systems are often nonlinear and why nonlinear systems can exhibit behavior that linear systems never can, including:

• Multiple equilibria
• Threshold effects
• Sudden transitions
• Bifurcations

These ideas are essential before moving toward stochastic differential equations.

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Key Takeaways

• A phase plane visualizes a system of differential equations.
• Each point represents a possible state of the system.
• Trajectories show how the system evolves over time.
• Vector fields show the direction of motion everywhere.
• Stable nodes attract trajectories.
• Unstable nodes repel trajectories.
• Saddle points attract in some directions and repel in others.
• Spiral points create oscillations.
• Eigenvalues determine the geometry of the phase plane.
• Phase planes provide intuition that equations alone often cannot.