Phase Portraits — Mathematics Knowledge System

Overview

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane (or higher-dimensional phase space). Instead of solving differential equations analytically, the phase portrait reveals the qualitative behavior of all solutions simultaneously: where they converge, diverge, spiral, or oscillate. Nullclines — curves where one derivative vanishes — divide the phase plane into regions of consistent sign, allowing one to sketch flow direction without integration. Equilibrium points (fixed points) appear where all derivatives vanish simultaneously, and the topology of trajectories near these points encodes the long-term fate of nearby solutions.

Intuition

Imagine a particle moving in the plane according to dx/dt = f(x,y), dy/dt = g(x,y). At every point (x,y), the vector (f,g) tells the particle which way to move next. Drawing these arrows densely gives the vector field; the curves that flow along these arrows — never crossing them, always tangent — are the trajectories. Nullclines are like invisible walls where the horizontal or vertical speed hits zero: the x-nullcline is where dx/dt=0 (so trajectories cross it vertically), and the y-nullcline where dy/dt=0 (trajectories cross it horizontally). Fixed points sit at intersections of x- and y-nullclines. By sketching nullclines and testing the sign of each derivative in each region, you can draw the phase portrait without solving a single integral.

Formal Definition

Definition

Consider an autonomous planar system. The phase portrait is the collection of all orbits in the phase plane, together with their orientation (direction of increasing time).

\dot{x} = f(x, y), \quad \dot{y} = g(x, y)

Autonomous planar system

\mathcal{N}_x = \{(x,y) : f(x,y) = 0\}, \quad \mathcal{N}_y = \{(x,y) : g(x,y) = 0\}

x- and y-nullclines

\text{Fixed points: } \{(x^*, y^*) : f(x^*,y^*) = 0 \text{ and } g(x^*,y^*) = 0\}

Equilibria (fixed points)

\frac{dy}{dx} = \frac{g(x,y)}{f(x,y)}

Slope of trajectory at (x,y)

Notation

Notation	Meaning
$\mathcal{N}_x$	Curve where dx/dt = 0; trajectories cross this vertically
$\mathcal{N}_y$	Curve where dy/dt = 0; trajectories cross this horizontally
$(x^, y^)$	Equilibrium where both derivatives vanish
$\gamma(t)$	Solution curve (x(t), y(t)) in phase space

Theorems

Theorem X: Theorem

See definition.

Theorem X: Theorem

See definition.

Worked Examples

1
Find x-nullcline: set dx/dt = 0, so x - y = 0, i.e., the line y = x.
$\mathcal{N}_x: y = x$
2
Find y-nullcline: set dy/dt = 0, so x + y = 0, i.e., the line y = -x.
$\mathcal{N}_y: y = -x$
3
Fixed point: both nullclines intersect at (0,0), which is the only equilibrium.
$(x^*, y^*) = (0, 0)$
4
Test sign in each region. In the region x > 0, y > 0, y < x (between nullclines): dx/dt = x-y > 0, dy/dt = x+y > 0. Arrows point right and up.
5
The Jacobian at (0,0) is [[1,-1],[1,1]] with eigenvalues 1 ± i, so (0,0) is an unstable spiral. Trajectories spiral outward.

✓ Answer

The phase portrait shows an unstable spiral at the origin, with trajectories winding outward counterclockwise.

Practice Problems

Easyfree response

Find the nullclines and fixed points of the system dx/dt = y, dy/dt = -sin(x).

Easyapplication

For dx/dt = x(2-x-y), dy/dt = y(3-x-2y), determine the fixed points in the first quadrant and describe the nullclines.

Common Mistakes

Common Mistake

Confusing x-nullclines and y-nullclines: students think the x-nullcline means x is the variable, so motion is horizontal.

On the x-nullcline, dx/dt = 0, so horizontal velocity is zero and motion is purely vertical. On the y-nullcline, dy/dt = 0, so motion is purely horizontal.

Common Mistake

Drawing trajectories that cross each other in the phase portrait.

By the uniqueness theorem, distinct trajectories can never intersect; if your sketch shows a crossing, at least one trajectory is drawn incorrectly.

Common Mistake

Drawing arrows along nullclines as if they indicate direction of flow.

On a nullcline, one velocity component is zero; the trajectory crosses the nullcline rather than flowing along it (unless it is a fixed point where both components vanish).

Common Mistake

Assuming the only fixed points are at the origin.

Fixed points occur at intersections of the x- and y-nullclines; a system can have many fixed points far from the origin.

Quiz

Historical Background

Henri Poincare pioneered the geometric study of differential equations in the 1880s, introducing the concept of phase space and the qualitative theory of ODEs in his landmark memoirs 'Sur les courbes definies par une equation differentielle.' This approach, which focused on the topology of solution curves rather than explicit formulas, was revolutionary. Poincare's work laid the groundwork for the entire field of dynamical systems, and the phase portrait became its primary visual language. In the early 20th century, Andronov and his school in the Soviet Union further developed these geometric methods, applying them to nonlinear oscillations in mechanics and electrical engineering.

1881-1886
Poincare publishes four memoirs on curves defined by differential equations, founding qualitative ODE theory
Henri Poincare
1920s
Andronov and colleagues apply phase plane methods to oscillation problems
Aleksandr Andronov
1960s
Phase portrait methods become standard in nonlinear dynamics textbooks
Jordan, Smith

Summary

A phase portrait displays all trajectories of an autonomous system simultaneously, giving global qualitative information without explicit solutions.
Nullclines (where one derivative vanishes) divide the phase plane into regions of consistent flow direction and always meet at fixed points.
The uniqueness theorem guarantees trajectories never cross, making the phase portrait a well-defined foliation of the plane minus the fixed points.
Phase portraits reveal long-term behavior — stable equilibria, unstable equilibria, limit cycles, separatrices — that is invisible in explicit solution formulas.

References

BookStrogatz, S. H. Nonlinear Dynamics and Chaos. Westview Press, 2015. Chapters 2-6.
WebsiteWikipedia — Phase portrait