The Phase Plane, Critical Points, and Stability

A nonlinear differential equation rarely has a closed-form solution. The alternative, created by Poincaré in the 1880s, is to abandon the search for explicit solutions and instead ask geometric questions: where does the system sit at rest, what happens to a state that starts near a rest position, and what does the family of all solution curves look like as a whole.¹ The setting for these questions is the phase plane, and the vocabulary is stability.

For the linear constant-coefficient system ${\mathbf{x}}^{\prime }=A\mathbf{x}$ every case can be worked out exactly from the eigenvalue analysis of systems. The catalogue of behaviors it produces is what the nonlinear theory reuses to classify equilibria.

Autonomous systems and the phase plane

We restrict attention to a system of two first-order equations in which the right-hand sides do not depend on $t$:

The geometric picture is possible only because the system is autonomous. Because the direction field $\mathbf{f}$ does not change with $t$, the trajectory through a point $(x_0,y_0)$ is the same regardless of when the solution passes through it. Assuming $F$ and $G$ are continuous with continuous first partial derivatives, the existence–uniqueness theorem guarantees exactly one solution through each point, so exactly one trajectory passes through each point of the plane.² Two consequences follow, and both are used constantly:

• Trajectories do not cross. If two trajectories met at a point, uniqueness would be violated there.
• A single phase portrait describes every solution at once. Time-translating a solution slides it along the same curve, so the picture of all curves is time-independent.

The state moves along its trajectory with velocity ${\mathbf{x}}^{\prime }(t)=\mathbf{f}(\mathbf{x})$; the direction field assigns that velocity vector to every point, and the trajectories are the curves everywhere tangent to it.

Critical points

The points where the velocity vanishes organize the picture.

For the linear system ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with $\det A\ne 0$, the only critical point is the origin. For a nonlinear system there may be many, each governing the trajectories in its neighborhood. A trajectory that is not itself an equilibrium can approach a critical point only as $t\to \infty$ or $t\to -\infty$; it never reaches one in finite time.³

Trajectories of the linear system

Take ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with $A$ a $2\times 2$ constant matrix and $\det A\ne 0$. Seeking solutions $\mathbf{x}=\mathbf{\xi }{e}^{rt}$ leads, as usual, to the eigenvalue problem $(A-rI)\mathbf{\xi }=0$, so the eigenvalues $r_1,r_2$ of $A$ and their eigenvectors govern everything. Five cases exhaust the possibilities, and each produces a distinct geometric type of critical point at the origin.

Case 1 — real eigenvalues, same sign (node). With $r_1<r_2<0$ the general solution is $\mathbf{x}=c_1{\mathbf{\xi }}^{(1)}{e}^{r_{1}t}+c_2{\mathbf{\xi }}^{(2)}{e}^{r_{2}t}$, and every trajectory tends to the origin as $t\to \infty$. Writing $\mathbf{x}=e^{r_{2}t}\left[c_1{\mathbf{\xi }}^{(1)}{e}^{(r_1-r_2)t}+c_2{\mathbf{\xi }}^{(2)}\right]$ shows that, because $r_1-r_2<0$, the bracket is dominated by its second term for large $t$: every trajectory except the pair along ${\mathbf{\xi }}^{(1)}$ enters the origin tangent to ${\mathbf{\xi }}^{(2)}$, the slow eigendirection. This is a node (a nodal sink when both eigenvalues are negative, a nodal source when both are positive, with the arrows reversed).

Case 2 — real eigenvalues, opposite sign (saddle). With $r_2<0<r_1$, the solutions along ${\mathbf{\xi }}^{(2)}$ approach the origin while those along ${\mathbf{\xi }}^{(1)}$ leave it, and a generic trajectory comes in from the ${\mathbf{\xi }}^{(2)}$ direction and departs asymptotic to ${\mathbf{\xi }}^{(1)}$. The origin is a saddle point, always unstable: the only trajectories that approach it are the two along the negative eigendirection.

Case 3 — repeated eigenvalue (proper or improper node). If $r_1=r_2=r$ has two independent eigenvectors, every trajectory is a ray through the origin (a proper node, or star point). If there is only one eigenvector $\mathbf{\xi }$, the solution carries a $t{e}^{rt}$ term and all trajectories enter tangent to $\mathbf{\xi }$; this is an improper (degenerate) node.

Case 4 — complex eigenvalues $\lambda \pm i\mu$, $\lambda \ne 0$ (spiral). In polar coordinates the system reduces to $r^{\prime }=\lambda r$, ${\theta }^{\prime }=-\mu$, so $r=c{e}^{\lambda t}$ and $\theta =-\mu t+{\theta }_0$. The radius grows or decays exponentially while the angle turns steadily: the trajectories are spirals, spiraling inward if $\lambda <0$ (spiral sink) and outward if $\lambda >0$ (spiral source).

Case 5 — pure imaginary eigenvalues $\pm i\mu$ (center). Now $\lambda =0$, so $r^{\prime }=0$: the radius is constant and every trajectory is a closed curve (a circle, or in general an ellipse) around the origin. The origin is a center, and every solution is periodic with period $2\pi /\mu$.

The table collects the classification and reads directly off the eigenvalues.

Stability, precisely

Three definitions classify the long-term behavior of solutions near a critical point. Stated for a general $n$-dimensional autonomous system ${\mathbf{x}}^{\prime }=\mathbf{f}(\mathbf{x})$, use $\Vert \mathbf{x}\Vert$ for the length of the vector $\mathbf{x}$.⁴

Stability is the statement that a state starting close stays close. It does not require the state to return to equilibrium; a center is stable, its orbits circling forever at fixed distance without approaching the middle.

Asymptotic stability is strictly stronger: nearby states must stay close and eventually return. The limit condition alone is not enough — one can construct systems in which every trajectory reaches ${\mathbf{x}}_0$ yet a trajectory first wanders arbitrarily far before returning, so ${\mathbf{x}}_0$ is not stable. Both conditions are required.

Reading the five cases against these definitions gives the master table for the linear system.⁴ Everything hinges on the sign of the real parts of the eigenvalues.

Every trajectory of a linear system, after a long time, does exactly one of three things: it approaches the origin, it traverses a closed orbit around the origin, or it becomes unbounded. These are asymptotic stability, stability, and instability respectively.

The trace–determinant plane

The eigenvalues of $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ are the roots of $r^2-pr+q=0$, where

so $r_{1,2}=\frac{1}{2}\left(p\pm \sqrt{p^2-4q}\right)$, and $r_1{r}_2=q$, $r_1+r_2=p$. The two numbers $p$ and $q$, together with the discriminant $\Delta =p^2-4q$, decide the type and stability without ever computing the eigenvalues explicitly:⁵

• $q<0$: the eigenvalues are real with opposite signs — a saddle point, always unstable.
• $q>0$, $\Delta >0$: real eigenvalues of the same sign — a node, stable if $p<0$ and unstable if $p>0$.
• $q>0$, $\Delta <0$: complex eigenvalues $\lambda \pm i\mu$ with $\lambda =p/2$ — a spiral, stable if $p<0$ and unstable if $p>0$.
• $p=0$, $q>0$: pure imaginary eigenvalues — a center, stable.
• $\Delta =0$ ($p^2=4q$): a repeated eigenvalue — the boundary parabola of proper and improper nodes.

The whole classification is a partition of the $(p,q)$ half-plane $q$-versus-$p$ by the axes and the parabola $q=p^2/4$.

The left half-plane $p<0$ is where trajectories decay; the right half-plane $p>0$ is where they grow. The positive $q$-axis, the single line $p=0$ with $q>0$, consists of centers separating stable spirals from unstable ones. This boundary is why the nonlinear theory is delicate: a center is the one case where an arbitrarily small change to the system can flip it either way.

Nullclines

For a system that is not linear, there is no eigenvalue shortcut to the global picture, but the direction field still has a skeleton. The nullclines are the curves where one component of the velocity vanishes.

Nullclines partition the plane into regions inside which the signs of $x^{\prime }$ and $y^{\prime }$ are constant, so the general direction of motion — up-right, up-left, and so on — is fixed within each region. Sketching the two nullclines, marking their intersections as critical points, and recording the sign of $(x^{\prime },y^{\prime })$ in each region gives a qualitatively correct phase portrait before any linearization.

Carryover from the linear case

Two features of the linear case are special and do not carry over to nonlinear systems:

• A linear system has a single critical point. A nonlinear system typically has several, each governing its own neighborhood.
• Linear stability is global. If the origin of a stable linear system attracts nearby trajectories, it attracts all of them. For a nonlinear system, a stable critical point attracts only trajectories that begin in its basin of attraction, and mapping that basin is a separate problem.

Zooming in on any critical point of a nonlinear system, the direction field comes to resemble that of a linear system, and the Jacobian at that point makes the resemblance precise.