Ordinary Differential Equations 16 | Periodic Solutions and Fixed Points [dark version]

A system of ordinary differential equations can produce three qualitatively different long-term behaviors—trajectories that never repeat, trajectories that loop periodically, and trajectories that stay put forever—and those behaviors can be read directly from the phase portrait. Fixed points occur exactly where the vector field vanishes: if V(x*) = 0, then the constant solution α(t) = x* satisfies α(t) = α(0) for all t, so the orbit is just a single point. Periodic solutions occur when the state repeats after some time T: α(t + T) = α(t) for every t, which forces the orbit to be a closed curve. In contrast, injective (non-repeating) solutions trace out orbits that do not hit the same point twice; among these, only the injective case may fail to exist for all time, meaning the maximal solution’s domain might be a proper subset of ℝ.

These distinctions connect tightly to how orbits behave under uniqueness: solutions to an initial value problem cannot cross in phase space. With a locally Lipschitz vector field V, each initial condition x0 generates a unique maximal solution α(t), and the “no-crossing” rule implies that if a trajectory ever revisits a point, it must be a fixed point or a periodic orbit. For periodic solutions, there is a minimal period (unlike fixed points, where any T works because the solution never changes). This classification—injective trajectories, fixed points, and periodic orbits—organizes the face portrait into a small set of geometric patterns.

To make the ideas concrete, the discussion turns to a pendulum-type equation: x¨ = −sin(x). Converting it to a first-order system by setting x1 = x and x2 = x˙ yields x1˙ = x2 and x2˙ = −sin(x1), so the vector field in ℝ² is V(x1, x2) = (x2, −sin(x1)). Instead of sketching V directly, a key trick identifies a conserved quantity. If there exists a function f(x1, x2) whose value stays constant along solutions, then trajectories must lie on contour lines of f.

The constancy condition is derived using the chain rule: requiring d/dt f(α(t)) = 0 for all t is equivalent to ∇f(α(t)) · α˙(t) = 0. Since α˙(t) = V(α(t)), this becomes the geometric requirement that ∇f is perpendicular to V everywhere. A suitable choice is f(x1, x2) = (1/2)x2² − cos(x1), whose gradient is (sin(x1), x2). This gradient is perpendicular to V, so f remains constant along trajectories. As a result, contour plots of f immediately reveal the phase portrait: fixed points occur where ∇f = 0, which forces x2 = 0 and sin(x1) = 0, giving x1 = kπ for integers k. The pendulum’s “normal” periodic motion appears as closed contour loops around the stable equilibrium points, while the equilibria at the top position (near the upside-down pendulum) also show up as fixed points—typically unstable even though they still satisfy the differential equation.

Finally, the conserved-energy picture explains the third category: trajectories with sufficiently high or low x2 correspond to motion that does not settle into the closed periodic loops, producing injective orbits that keep moving without repeating. The fixed points can also be stable or unstable depending on what happens when initial conditions are slightly perturbed, a topic saved for the next step via local linearization.