What Is a Phase Line?
A phase line is a one-dimensional diagram that captures the entire long-term behavior of an autonomous differential equation, one whose rate depends only on the current value, not on time:
y′ = f(y)
The equilibria are the values y* where f(y*) = 0, at which the system holds steady. Between them, the sign of f tells you which way solutions move: where f > 0, y increases (arrow up); where f < 0, y decreases (arrow down). Marking equilibria as points and the flow direction as arrows on a vertical line produces the phase line, from which you can read the stability of each equilibrium without ever solving the equation. A stable equilibrium (a sink) has arrows pointing toward it from both sides; an unstable one (a source) has arrows pointing away; a semistable one is approached from one side and fled on the other. This calculator finds the equilibria, classifies each by the sign of f around it, and draws solution curves flowing toward or away from the equilibrium levels.
How to Use the Phase Line Calculator
Enter the right-hand side f(y) (depending only on y, since the equation is autonomous) and a window of y-values to analyze. The steps find the equilibria (roots of f), classify each as stable, unstable, or semistable from the sign of f just below and above it, and describe the flow. The graph shows a family of solution curves over time, converging to stable equilibria and diverging from unstable ones, with equilibrium levels drawn as horizontal lines. This is the qualitative-analysis companion to the numerical slope field generator and Euler's method calculator; the two-dimensional analog is the phase portrait of a linear system.
Worked Example
The logistic equation y′ = y(1 − y) is the definitive phase-line example. Its equilibria are where y(1 − y) = 0, namely y = 0 and y = 1. Checking the sign of f in each region:
for y < 0: f < 0 (down); for 0 < y < 1: f > 0 (up); for y > 1: f < 0 (down)
Around y = 0, the arrows point away on the upper side (up) and away on the lower side (down): y = 0 is unstable, a source. Around y = 1, the arrows point toward it from below (up) and from above (down): y = 1 is stable, a sink. This is exactly the biology of logistic population growth: the extinction state y = 0 is unstable (any small population grows away from it), while the carrying capacity y = 1 is stable (populations above or below it settle toward it). Without solving the differential equation, the phase line has told us the complete fate of every solution: any positive starting population approaches the carrying capacity, and the origin repels. The semistable chip f = y² shows the third type: f ≥ 0 everywhere, so solutions approach y = 0 from below but flee it above, a half-stable equilibrium where the derivative test is inconclusive.
Why the Sign of f Determines Stability
Phase-line analysis works because autonomous equations have a special structure: the rate of change is a fixed function of position, so the direction of motion at any value y is completely determined by the sign of f(y), independent of time. This lets you predict the future from the sign pattern alone. At an equilibrium y*, stability is decided by how f behaves nearby, and there is a clean derivative test: if f′(y*) < 0, the equilibrium is stable (f is decreasing through zero, so it is positive just below y* pushing up and negative just above pushing down, both toward y*); if f′(y*) > 0, it is unstable (f increasing through zero pushes away on both sides); and if f′(y*) = 0, the test is inconclusive and you must examine the sign of f directly, which is where semistable equilibria appear.
Common Mistakes to Avoid
- Using it on non-autonomous equations. The phase line requires y′ = f(y) with no explicit t-dependence. If f depends on t, the flow direction changes over time and a single phase line cannot capture the behavior.
- Reading the derivative test backward. Stable means f′(y*) < 0 (f decreasing through the equilibrium), unstable means f′(y*) > 0. Inverting this misclassifies sinks as sources.
- Ignoring semistable cases. When f touches zero without changing sign (a double root, like y²), the equilibrium is semistable, attracting on one side and repelling on the other. The linear derivative test gives zero and misses this; check the sign directly.
- Confusing the phase line with solution graphs. The phase line lives in the y-axis (state space); solution curves live in the t-y plane. The phase line summarizes where solutions go, not the time course of getting there.
- Missing equilibria. Every root of f is an equilibrium; missing one (especially a repeated or closely spaced root) leaves gaps in the stability picture. Find all roots in the region of interest.
Real-World Applications
Phase-line analysis is the primary tool for understanding the qualitative behavior of any system modeled by an autonomous first-order equation, and such models are everywhere. In population biology, the logistic equation and its variants describe how populations approach carrying capacity, and phase-line analysis identifies stable population levels, extinction thresholds, and the tipping points (unstable equilibria) that separate different long-term outcomes, crucial for conservation and fisheries management, where harvesting can shift or destroy a stable equilibrium. Epidemiology uses phase lines to find endemic equilibria and the basic reproduction number's threshold.
Frequently Asked Questions
What is an autonomous differential equation?
One whose right-hand side depends only on the dependent variable, y′ = f(y), with no explicit time dependence. This means the rate of change is fixed by the current value alone, which is exactly what makes phase-line analysis possible: the flow direction at each value never changes over time.
What are equilibria and how do I find them?
Equilibria are constant solutions where y′ = 0, so they are the roots of f(y) = 0. At an equilibrium the system holds steady forever. Finding all roots of f in the region of interest gives the complete set of equilibrium values to classify for stability.
How do I classify an equilibrium's stability?
By the sign of f around it, or the derivative test: f′(y*) < 0 means stable (a sink, arrows point in), f′(y*) > 0 means unstable (a source, arrows point out), and f′(y*) = 0 is inconclusive, requiring direct sign analysis that may reveal a semistable equilibrium.
What is a semistable equilibrium?
One that attracts solutions from one side but repels them on the other, occurring when f touches zero without changing sign (a double root, like f = y² at y = 0). Solutions approach from the attracting side and flee on the repelling side, a half-stable behavior the linear derivative test cannot detect.
Why does the phase line predict behavior without solving the equation?
Because the sign of f at each value determines the direction of motion there, and for an autonomous equation that direction is fixed in time. Tracking where solutions must flow, up where f > 0, down where f < 0, toward or away from equilibria, gives the complete long-term picture from sign information alone.
How does this relate to the logistic model?
The logistic equation y′ = y(1 − y) has an unstable equilibrium at 0 (extinction) and a stable one at 1 (carrying capacity). The phase line shows every positive population approaching the carrying capacity, which is the model's central prediction, obtained purely from the sign structure without solving.
Can I use a phase line if f depends on time?
No. Time dependence makes the equation non-autonomous, so the flow direction at a given value changes over time and cannot be summarized by a single static diagram. Non-autonomous equations require other tools, though some can be analyzed with extended phase-space or Poincaré methods.
What is the connection to eigenvalue stability analysis?
For a one-dimensional equation, f′(y*) plays the role of the eigenvalue: negative means attracting (stable), positive means repelling (unstable). This is the one-dimensional case of the linearization used for systems, where the eigenvalues of the Jacobian matrix determine stability of equilibria in the phase plane.
What do the solution curves on the graph show?
They show y versus time for several starting values, illustrating the flow: curves converge to stable equilibria (horizontal asymptotes at sinks) and diverge from unstable ones. The phase line summarizes these curves' destinations, and the time plot shows the dynamics of reaching them.
Why are unstable equilibria important if solutions never stay there?
Because they are thresholds separating different outcomes. An unstable equilibrium is a tipping point: starting just above versus just below it leads to entirely different fates. In population, disease, and climate models, these thresholds mark critical boundaries, making unstable equilibria as important to identify as stable ones.