Flows and Fixed Points

A dynamical system is any rule that tells a quantity how to change from one instant to the next. The simplest and most fundamental version is a single quantity x(t) whose rate of change depends only on its current value:

\dot{x} = f(x), \qquad \dot{x} \equiv \frac{dx}{dt}.

This is a first-order autonomous ODE — "autonomous" because t never appears on the right, only x. You met the machinery for solving such equations when you studied systems of ODEs. Here we change the question entirely. Instead of grinding out a formula for x(t), we ask the geometer's question: without solving anything, what does the motion look like? Does it rush off to infinity, settle to rest, or hover forever? That shift — from formulas to qualitative geometry — is the whole spirit of dynamical systems.

The flow on the line

Picture the x-axis as a river. At each point x the value f(x) is the velocity of the current there: where f(x) > 0 the water flows right (x increases), where f(x) < 0 it flows left. A tiny cork dropped anywhere is simply carried along by the current. This picture of "drop a particle and let it drift" is called the flow, and the arrows it induces on the line are the phase line.

The special places are where the current vanishes — where f(x^\*) = 0. A cork placed exactly there never moves. These are the fixed points (also called equilibria, steady states, or critical points): the constant solutions x(t) \equiv x^\* of the ODE. Finding them is pure algebra — solve f(x) = 0 — and everything about the long-term behaviour hangs on them.

You can read stability off the graph of f at a glance: at a fixed point the curve crosses the axis, and a downward crossing (slope negative) is stable, an upward crossing (slope positive) is unstable. This is the germ of the derivative test we sharpen in the next lesson.

See the current: \dot{x} = x - x^3

Take f(x) = x - x^3 = x(1-x)(1+x). Its zeros — the fixed points — are x^\* = -1,\ 0,\ +1. The graph below is f itself. Read the flow straight off it: where the curve sits above the axis the current runs right; where it dips below, the current runs left.

Trace the arrows. Between -1 and 0 the curve is below the axis, so the flow is leftward — towards -1, away from 0. Between 0 and 1 the curve is above, so the flow is rightward — again away from 0, towards 1. So x^\* = \pm 1 are stable (downward crossings) and x^\* = 0 is unstable (upward crossing). Any cork not started exactly at the origin eventually drifts to +1 or -1. We have described the fate of every initial condition without solving a single differential equation.

On the line, a trajectory is monotonic: it can only increase, only decrease, or sit still. To come back and repeat a value it would have to reverse direction, but the direction at each point is fixed once and for all by the sign of f — the current at a spot never changes its mind. So no oscillations, no periodic orbits, no overshoot are possible for \dot{x} = f(x). Every bounded trajectory simply slides monotonically onto a fixed point. To get genuine oscillation you need at least two dimensions — a plane where a trajectory can loop around without crossing itself. That is exactly why the phase plane, two lessons ahead, is such a leap.

Worked example: a population with a threshold

A colony of size N \ge 0 grows logistically but is culled at a constant rate h:

\dot{N} = N\left(1 - \frac{N}{K}\right) - h.

Step 1 — fixed points. Set the right side to zero. With K = 4 and h = 3/4 this is N - \tfrac{N^2}{4} - \tfrac34 = 0, i.e. N^2 - 4N + 3 = 0, so

N^\* = 1 \quad\text{or}\quad N^\* = 3.

Step 2 — sign of f between them. The parabola f(N) = -\tfrac14(N-1)(N-3) opens downward, so f < 0 for N < 1, f > 0 for 1 < N < 3, and f < 0 again for N > 3.

Step 3 — classify. At N^\* = 1 the flow goes from left (below) to right (above): the arrows point away — unstable. At N^\* = 3 the flow goes from right to left: arrows point towards — stable. The biological reading is stark: the lower equilibrium is a survival threshold. Start above N = 1 and the colony climbs to the healthy carrying level N = 3; start even slightly below and harvesting wins — the colony crashes to extinction. A single unstable fixed point is drawing the line between life and death, and we found it with nothing but a quadratic.

Existence, uniqueness, and blow-up

Two quiet facts keep the phase-line picture honest. First, if f is continuously differentiable, the existence–uniqueness theorem guarantees exactly one trajectory through each starting point — so trajectories never cross, and a cork approaching a stable fixed point can only approach it, never arrive in finite time (it would have to collide with the constant solution sitting there).

Second, "reaching" a fixed point is asymptotic, but escaping to infinity can be brutally fast. For \dot{x} = x^2 with x(0) = 1, separating variables gives x(t) = 1/(1-t) — the solution blows up at t = 1, escaping to infinity in finite time. Bounded flows limp onto equilibria; unbounded ones can detonate.

A fixed point is not "where the graph of x(t) is flat" — it is where the graph of f(x) crosses zero. Keep the two pictures apart: the phase line lives in x-space (with time hidden), while x(t) is the time series. And do not confuse the value f(x^\*) = 0 with the slope f'(x^\*): it's f = 0 that makes a point fixed, and the sign of the slope f' that makes it stable or unstable. A very common slip is to declare a point stable just because f is small nearby — smallness is irrelevant; only the direction the arrows point matters.