Phase Portraits

Beyond Individual Solutions

In the previous chapter, we introduced systems of differential equations and the phase plane. We saw that the vector field shows us the direction of motion at every point, and that solution curves, or trajectories, flow through this field like leaves on a river.

Now we take a step back and ask a bigger question: what is the complete picture of a system's behavior? Not just one trajectory, but all of them. Not just where a particular solution goes, but the overall geometry of the flow.

This complete picture is called a phase portrait. It is the most powerful tool we have for understanding dynamical systems qualitatively.

What Is a Phase Portrait?

A phase portrait is a visualization of all possible trajectories of a two-dimensional system

in the phase plane. It shows:

• The vector field: small arrows indicating the direction of motion at each point
• Trajectories: curves traced out by solutions as time evolves
• Equilibrium points: where the vector field vanishes and trajectories begin, end, or circulate
• Special structures: separatrices, limit cycles, and other organizing features

The phase portrait reveals the qualitative behavior of the system. We can read off whether solutions grow, decay, oscillate, or approach equilibria, all without solving a single equation explicitly.

Each equilibrium type creates a distinctive pattern in the phase portrait:

• Saddle points have trajectories approaching along one direction and fleeing along another, creating a characteristic X pattern
• Nodes attract (stable) or repel (unstable) trajectories from all directions
• Spirals wind inward (stable) or outward (unstable) around the equilibrium
• Centers have trajectories forming closed loops that neither approach nor recede

The phase portrait captures the essence of the system's dynamics at a glance.

Equilibrium Points

The most important features of a phase portrait are the equilibrium points (also called fixed points, critical points, or stationary points). These are points $(x^*, y^*)$ where both derivatives vanish:

At an equilibrium, the vector field has zero magnitude. A trajectory that starts exactly at an equilibrium stays there forever. The constant function $(x(t), y(t)) = (x^*, y^*)$ is a solution.

But equilibria are not just static points. They organize the entire phase portrait. Nearby trajectories either approach the equilibrium, recede from it, or circulate around it. Understanding the local behavior near equilibria tells us a great deal about the global structure of the system.

Finding equilibria requires solving the simultaneous equations $f(x, y) = 0$ and $g(x, y) = 0$. For nonlinear systems, there may be multiple equilibria, each with its own local behavior.

Nullclines: Where Motion Is Constrained

How do we find equilibria and understand the flow in between? The key is nullclines.

An x-nullcline is the set of points where $x' = 0$. Along this curve, the vector field is purely vertical: motion is only in the $y$ direction.

A y-nullcline is the set of points where $y' = 0$. Along this curve, the vector field is purely horizontal: motion is only in the $x$ direction.

Equilibrium points occur exactly where nullclines intersect. At such points, both $x' = 0$ and $y' = 0$, so there is no motion at all.

Adjust the parameters and watch how the nullclines move. When they intersect, that intersection is an equilibrium. The angle and position of intersection determine what kind of equilibrium it is.

Nullclines divide the phase plane into regions. In each region, the signs of $x'$ and $y'$ are constant. This means trajectories in that region always move in a consistent direction: perhaps up and to the right, or down and to the left. The nullclines themselves are the boundaries where trajectories switch from moving right to moving left (at the x-nullcline) or from moving up to moving down (at the y-nullcline).

The Geometry of Nullclines

Consider what happens when a trajectory crosses a nullcline:

• Crossing the x-nullcline (where $x' = 0$): the trajectory is momentarily vertical, changing from moving right to moving left or vice versa
• Crossing the y-nullcline (where $y' = 0$): the trajectory is momentarily horizontal, changing from moving up to moving down or vice versa

This means trajectories can only enter or exit regions by crossing nullclines at specific angles. The nullclines act like guardrails, channeling the flow.

In this predator-prey style system, the nullclines create four regions. In each region, the signs tell us the direction of flow. Notice how the trajectory is funneled by the nullcline structure, spiraling around the equilibrium at $(1, 1)$.

This is the power of nullcline analysis: without solving the equations, we can sketch the qualitative behavior of trajectories by understanding which direction they flow in each region and how they must cross from one region to another.

Trajectories and Orbits

A trajectory (or orbit) is the path traced by a solution $(x(t), y(t))$ as $t$ varies. Each initial condition generates a unique trajectory.

Trajectories have several important properties:

Uniqueness: Trajectories never cross. If two solutions passed through the same point at different times, uniqueness of solutions would give them the same future and past, so they would be the same trajectory.

Direction: Trajectories have a direction of travel, indicated by arrows. Time flows forward along the trajectory.

Continuity: Trajectories are smooth curves that depend continuously on initial conditions (at least for well-behaved systems).

Limiting behavior: As $t \to \infty$, trajectories approach attracting equilibria, limit cycles, or escape to infinity. As $t \to -\infty$, they emerge from repelling equilibria, limit cycles, or infinity.

Click to place initial conditions and watch trajectories unfold. Notice how:

• For a stable spiral, all trajectories wind inward toward the equilibrium
• For a saddle, some trajectories approach and others flee
• For a center, trajectories form closed orbits that neither approach nor recede

The family of all trajectories fills the phase plane, and together they form the phase portrait.

Separatrices

In saddle point systems, certain special trajectories divide the phase plane into regions with different fates. These are called separatrices.

For a saddle at the origin, there are typically four separatrices: two approaching the saddle (along the stable direction) and two leaving it (along the unstable direction). Trajectories on different sides of a separatrix have qualitatively different long-term behavior.

Separatrices are the boundaries between basins of attraction. If a system has multiple equilibria, separatrices determine which initial conditions lead to which equilibrium.

Think of separatrices as watersheds. Rain falling on one side of a ridge flows to one river; rain on the other side flows elsewhere. Initial conditions on different sides of a separatrix evolve toward different fates.

Limit Cycles

Not all long-term behavior involves equilibrium points. Some systems exhibit limit cycles: isolated closed trajectories that are approached by neighboring trajectories.

A limit cycle is not just any closed orbit. It is isolated, meaning there are no other closed orbits nearby. Trajectories starting inside the limit cycle spiral outward toward it; trajectories starting outside spiral inward toward it. The limit cycle is an attractor for all nearby initial conditions.

Limit cycles represent sustained oscillation. Unlike the closed orbits around a center (which form a continuous family), a limit cycle is a unique periodic solution that the system settles into regardless of where it starts.

The Van der Pol oscillator is a famous example from electronics and biology. The system has a single unstable equilibrium at the origin, surrounded by a stable limit cycle. No matter where you start (except exactly at the origin), the trajectory eventually settles onto the same periodic motion.

As you increase the parameter $\mu$, the limit cycle becomes more pronounced. Small $\mu$ gives nearly sinusoidal oscillation; large $\mu$ produces relaxation oscillations with rapid transitions between slow phases.

Limit cycles appear in:

• Heartbeat rhythms
• Predator-prey population cycles
• Electrical oscillators
• Neural firing patterns
• Chemical reactions

Anywhere you see sustained, self-reinforcing oscillation, there is likely a limit cycle.

Reading a Phase Portrait

Given a phase portrait, here is how to extract information:

1. Locate equilibria. These are the points where trajectories converge, diverge, or circulate.

2. Classify each equilibrium. Is it a node, saddle, spiral, or center? Is it stable (attracting) or unstable (repelling)?

3. Identify nullclines. Where is the vector field purely horizontal or vertical?

4. Determine flow direction in each region. The signs of $x'$ and $y'$ tell you whether trajectories move up/down and left/right.

5. Look for separatrices. These divide regions with different long-term fates.

6. Check for limit cycles. Are there isolated closed orbits?

7. Trace typical trajectories. Where do they come from? Where do they go?

With practice, you can sketch a qualitative phase portrait from the equations alone, predicting behavior before any numerical simulation.

Linear vs. Nonlinear

For linear systems $\mathbf{x}' = A\mathbf{x}$, the phase portrait is determined entirely by the eigenvalues of $A$. We will explore this connection in the next chapter.

Nonlinear systems are richer. They can have multiple equilibria, limit cycles, and more exotic structures. Near each equilibrium, the behavior resembles that of a linear system (the linearization), but the global picture requires more care.

The beauty of phase portraits is that they work for both. Whether the system is linear or nonlinear, the geometric picture tells the story.

Building Intuition

Phase portraits are geometric. They reward visualization and exploration more than algebraic manipulation.

Here are practices that build intuition:

• Sketch by hand. Given a system, draw the nullclines, mark equilibria, determine flow directions, and sketch a few trajectories.
• Vary parameters. Watch how the phase portrait changes as you adjust coefficients. Small changes can cause qualitative shifts (bifurcations).
• Think in terms of flow. Imagine the phase plane as a fluid. Where does the fluid flow? Where does it collect or disperse?
• Connect to physical systems. A mass-spring system, a predator-prey model, a chemical reaction: each has a phase portrait that encodes its dynamics.

The more phase portraits you see and sketch, the more readily you will recognize patterns and predict behavior.

Key Takeaways

• A phase portrait is the complete picture of all trajectories in the phase plane, revealing the qualitative dynamics of a system
• Equilibrium points are where $f(x,y) = 0$ and $g(x,y) = 0$ simultaneously; they organize the entire flow
• Nullclines are curves where one component of velocity is zero; equilibria occur where nullclines intersect
• Nullclines divide the plane into regions where the flow direction is consistent; trajectories change direction only when crossing nullclines
• Trajectories never cross each other; they fill the phase plane and flow continuously from past to future
• Separatrices are special trajectories that divide the plane into regions with different long-term fates
• Limit cycles are isolated closed orbits that attract nearby trajectories, representing sustained periodic behavior
• The phase portrait can be read qualitatively without solving the equations: locate equilibria, draw nullclines, determine flow directions, and sketch trajectories