Phase Plane

A rigorous introduction to planar autonomous systems, trajectories, equilibria, linearization, and nullclines.

A phase plane shows how two state variables change together without using time as an axis. Small arrows indicate the local direction of change, two trajectories follow them, and an equilibrium point is marked where both derivatives vanish.

Precise definition

For an autonomous system $x'=f(x,y)$, $y'=g(x,y)$, the phase plane displays trajectories $(x(t),y(t))$ without using time as an axis. Equilibria satisfy $f(x_,y_)=g(x_,y_)=0$. The vector field $(f,g)$ is tangent to each oriented trajectory.

Notation and mathematical language

Nullclines are curves $f=0$ and $g=0$ where one component of motion vanishes. Linearization uses the Jacobian $J=\begin{pmatrix}f_x&f_y\\g_x&g_y\end{pmatrix}$ at an equilibrium. Eigenvalues classify hyperbolic linear behaviour: opposite signs give a saddle; both negative real parts a sink; both positive a source.

Conceptual picture

A phase trajectory records state combinations, while time determines how quickly it is traversed. Arrows show orientation. Closed curves can represent periodic motion; spirals combine rotation with attraction or repulsion; nullclines partition regions by horizontal and vertical direction.

Fully worked example

Interpretation and application

Phase planes model interacting species, chemical reactions, circuits, and mechanical position–velocity states. A phase portrait organizes possible dynamics but does not establish which variables causally drive each other unless the model and study design support that interpretation.