4.1 Linear Phase Portraits

The simplest system of two equations is one whose equations ignore each other:

$x^{'} = x, y^{'} = - y .$

Each unknown solves on its own: $x (t) = x_{0} e^{t}$ grows, $y (t) = y_{0} e^{- t}$ decays. The two rates, $plus 1$ and $minus 1$ , are the system's eigenvalues, and the directions along which the motion is pure exponential growth or pure exponential decay — here, the coordinate axes — are its eigendirections. A state on the positive $x$ -axis flees the origin along it forever; a state on the $y$ -axis slides down it toward the origin forever. Every other state feels both influences at once.

What does "both at once" look like? A solution is a moving point $(x (t), y (t))$ , and eliminating time exposes its path: the product $x (t) y (t) = x_{0} y_{0} e^{t} e^{- t} = x_{0} y_{0}$ never changes. Every orbit lies on a hyperbola $x y = c$ (or, when $c equals 0$ , on an axis). The figure draws three of them.

Three hyperbola-shaped arcs in the plane, viewed face-on: a red and a green arc sweeping through the upper-right quadrant toward the horizontal axis, and a blue arc mirrored into the upper-left quadrant. Each arc rides in along the vertical axis direction and out along the horizontal axis direction, the signature flow of a saddle point at the origin. — Three orbits of the saddle system $x^{'} = x$ , $y^{'} = - y$ , viewed face-on, each drawn for $minus 2 less than or equal to t less than or equal to 2$ : red through $open parenthesis 1, 2 close parenthesis$ , green through $open parenthesis 2, 1 close parenthesis$ , and blue through $open parenthesis minus 1, 1 close parenthesis$ . Each arc sweeps in along the vertical direction and out along the horizontal — the axes are the eigendirections, the only straight-line orbits.

Explore

Observe. Follow the red arc in the direction of increasing time. At $t = - 2$ it sits at $open parenthesis 0.14, 14.8 close parenthesis$ , high against the vertical axis; by $t equals 2$ it has swept down and out to $open parenthesis 7.4, 0.27 close parenthesis$ , flat along the horizontal. It falls in toward the origin, turns, and flees — and never actually arrives anywhere.
Predict. The red orbit passes through $open parenthesis 1, 2 close parenthesis$ and the green through $open parenthesis 2, 1 close parenthesis$ . Compute $x y$ for each point before looking closely: are these two different hyperbolas, or two pieces of the same one?
Verify. Both products equal $2$ — one hyperbola, $x y equals 2$ . Check the shared point $open parenthesis 2, 1 close parenthesis$ : the red solution reaches it at $t = \ln 2 \approx 0.69$ , while the green solution starts there at $t equals 0$ . Two different solutions, one orbit.
Observe. The blue arc, through $open parenthesis minus 1, 1 close parenthesis$ , rides the branch $x y = - 1$ in the second quadrant — the same shape, mirrored. Notice that no arc ever touches an axis: on the vertical axis $x equals 0$ forces $x prime equals 0$ , so a state born on an axis stays on it forever. The eigendirections are invariant lines, and other orbits can only approach them.
Predict, then reason. Suppose both equations drained: $x^{'} = - x$ , $y^{'} = - y$ . No figure needed — what quantity is now constant along orbits in place of $x y$ , what shape are the orbits, and where does every state end up?

Connect

The example generalizes further than it looks. A general linear system $x^{'} = a x + b y$ , $y^{'} = c x + d y$ hides its rates, but they are still there: along each eigendirection the motion is a pure exponential $e^{λ t}$ , and the pair of eigenvalues classifies the portrait completely. Real eigenvalues of opposite sign give a saddle — this figure: in along one direction, out along the other, hyperbolic arcs between. Real eigenvalues of the same sign give a node: every orbit sweeps into the origin (both negative) or out of it (both positive), with no way around. Complex eigenvalues $α \pm i β$ give a spiral — rotation at rate $β$ with amplitude $e^{α t}$ ; the phase loop of Chapter 2 was exactly this, eigenvalues $- a \pm i b$ . And purely imaginary eigenvalues give a center, the closed ellipses of the undamped spring. It is the damping-regime trichotomy again, wearing geometric clothes.

The portrait forgets the clock, and does so deliberately. Because the system is autonomous — time appears nowhere on the right-hand sides — delaying a solution produces another solution, and both trace the same curve. That is what the Explore steps uncovered: the green solution is the red one delayed by $ln 2$ . An orbit is a road; a solution is a journey along it with a schedule. The phase portrait is the road map — every road, no schedules — and that economy is why one still image can hold every possible history of the system at once.

An original work of XYZ Homework, built around interactive XYZ 3D figures, following the standard first-course sequence (first-order equations through systems and the phase plane). Distinct from the LibreTexts-sourced Differential Equations for Engineers (Lebl) edition, which has its own attribution. License: CC-BY-NC-SA-4.0.