Phase Plane

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.

Conditions and key results

Linearization reliably preserves local type for hyperbolic equilibria—none of the Jacobian eigenvalues has zero real part. Purely imaginary or zero-real-part cases are inconclusive for nonlinear systems. Crossing trajectories would violate uniqueness where the vector field is locally Lipschitz.

A reliable strategy

1. Find both nullclines and all their intersections to locate equilibria.
2. Evaluate sign pairs $(f,g)$ in the regions cut by nullclines and sketch representative arrows.
3. Compute the Jacobian and eigenvalues at each equilibrium, noting whether the hyperbolicity condition holds.
4. Combine local analysis with invariant sets or trajectories and verify orientation from the original system.

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.

Common mistakes

Verification and reasonableness

• Substitute equilibrium coordinates into both right-hand sides.
• Check eigenvalue sum and product against trace and determinant.
• Evaluate the vector field at sample points to confirm arrow orientation and region signs.

Practice

1. What equations define equilibria?
2. What eigenvalue signs define a saddle?
3. Does the phase plane display time directly?

Further deduction

For a $2\times2$ matrix, trace $\tau$ and determinant $\Delta$ give a quick map: eigenvalues solve $\lambda^2-\tau\lambda+\Delta=0$. If $\Delta<0$, the equilibrium is a saddle. If $\Delta>0$ and $\tau<0$, it is locally attracting when hyperbolic; $\tau>0$ gives repelling behaviour. The discriminant $\tau^2-4\Delta$ separates real nodes from complex spirals for the linear system.

An invariant curve can organize global behaviour. If a function $H(x,y)$ satisfies $\nabla H\cdot(f,g)=0$, then $H$ is constant along trajectories. For the undamped oscillator $x'=y,y'=-x$, $H=(x^2+y^2)/2$ is conserved, so trajectories are circles. This exact invariant proves closed orbits where linear arrow sketches only suggest them.

Related topics

• Systems of Differential Equations
• Equilibrium Solutions
• Eigenvalues