Not every system of differential equations needs to be solved explicitly to be understood. In this section, we focus on qualitative methods: tools that help us visualize how a system behaves, predict long-term motion, and recognize patterns like decay, spirals, and saddles without finding exact formulas. These ideas give us a big-picture view of linear systems and prepare us for deeper analysis later.
The phase plane is a two-dimensional space where we plot one variable on the horizontal axis and the other on the vertical axis. A solution to the system becomes a trajectory — a path through the plane traced as time flows.
To see the “push” of the system without fully solving it, we can draw a direction field (or slope field): at each point \((x, y)\text{,}\) we sketch a small arrow showing the vector \((dx/dt, dy/dt)\text{.}\)
In the phase plane, every trajectory moves toward the origin — trajectories are generally curved (except along the axes, which are straight-line solutions), since \(x\) and \(y\) decay independently at different rates.
For more interesting systems — like ones with partial or full coupling — the direction field can show curved paths, spirals, or saddle-shaped flows. This visual approach helps us predict system behavior even before we dive into equations.
This interactive slope field shows the direction of motion for the system \(x' = x + y\text{,}\)\(y' = -x + y\text{.}\) Each arrow represents the vector \((dx/dt, dy/dt)\) at that location.
An interactive figure pairing a phase-plane direction field on the left with a time plot on the right, so the trajectory in the x-y plane can be compared with the curves of x of t and y of t against time t.
Some systems push every solution toward a single point (stable equilibrium). Others send trajectories outward (unstable). Some cause spirals, as if the solution is both rotating and growing or shrinking at the same time.
These “big-picture” behaviors are not accidents — they are completely determined by the algebra of the system’s coefficients. The next subsection makes that connection precise using the eigenvalues from Solving Linear Systems.
SubsectionConnecting Eigenvalues to Phase Portraits
In Solving Linear Systems, we saw that the general solution of \(\vec{X}' = A\vec{X}\) is built from terms like \(\vec{v}e^{rt}\text{,}\) where \(r\) is an eigenvalue of \(A\text{.}\) Those eigenvalues are exactly what the phase portrait is showing us: each exponential rate controls whether trajectories grow, decay, or rotate. So once you know the eigenvalues, you can classify the behavior — usually without any plotting.
For a \(2\times 2\) system with distinct eigenvalues \(r_1\) and \(r_2\) (real or complex), the standard cases are: 14
The borderline case of a repeated eigenvalue (\(r_1 = r_2\)) is more delicate — there the eigenvectors also matter, and the portrait can be a star or a degenerate node. We won’t need that case in this book.
drawn in along one eigenvector, flung out along the other
complex, \(r = a \pm bi\) with \(a \lt 0\)
spiral sink
trajectories loop inward as they decay
complex, \(r = a \pm bi\) with \(a \gt 0\)
spiral source
trajectories loop outward as they grow
purely imaginary, \(r = \pm bi\)
center
trajectories orbit the origin in closed loops
The reading is straightforward: the real part of each eigenvalue controls growth (\(e^{at}\) grows when \(a \gt 0\text{,}\) decays when \(a \lt 0\)), and a nonzero imaginary part contributes rotation (through the \(\cos(bt)\) and \(\sin(bt)\) factors in the real form of the solution).
Let’s revisit the two systems we solved earlier through this lens:
In 🌌 Example 304, the eigenvalues were \(r_1 = 3\) and \(r_2 = -1\) — real with opposite signs. The phase portrait is a saddle: solutions starting along the eigenvector \(\begin{bmatrix}1\\-2\end{bmatrix}\) decay toward the origin, while every other solution is eventually carried off in the direction of \(\begin{bmatrix}1\\2\end{bmatrix}\) by the dominant \(e^{3t}\) term.
In 🌌 Example 305, the eigenvalues were \(r = 2 \pm 5i\) — complex with positive real part. The phase portrait is a spiral source: the \(e^{2t}\) factor pushes trajectories outward while the \(\cos(5t)\) and \(\sin(5t)\) factors rotate them.
This connection runs both ways. Eigenvalues predict the picture, and a glance at a phase portrait tells you the character of the eigenvalues — spirals mean complex eigenvalues, saddles mean real eigenvalues of opposite signs, and so on. You’ll practice both directions in the exercises.
