10.1 Orbits, equilibria and invariant sets
Definition 10.1.1. The orbit through \(x_0\) is the curve \(\{\phi _t(x_0)\}\) traced by the maximal solution. A point \(x^*\) with \(f(x^*) = 0\) is an equilibrium. A set \(S\) is invariant if \(\phi _t(S) \subseteq S\) for every \(t\) for which the flow is defined, and positively invariant if this holds for \(t \geq 0\). In the plane, the nullclines are the curves on which \(x_1' = 0\) or \(x_2' = 0\).
Lemma 10.1.2. Distinct orbits do not intersect. An orbit that is not an equilibrium is either a one-to-one image of an interval or a closed curve, and in the second case the solution is periodic.
Proof. If two orbits share a point \(z = \phi _s(x_0) = \phi _u(y_0)\), then \(\phi _t(y_0) = \phi _{t - u + s}(x_0)\) for all \(t\), by the flow property and uniqueness, so they are the same orbit. If \(\phi _{t_1}(x_0) = \phi _{t_2}(x_0)\) with \(t_1 < t_2\), then by the same argument \(\phi _{t + T}(x_0) = \phi _t(x_0)\) for every \(t\) with \(T = t_2 - t_1\), so the solution is periodic and the orbit closed; otherwise \(t \mapsto \phi _t(x_0)\) is one-to-one. □
The phase portrait is therefore a family of non-crossing curves, with equilibria as points and periodic orbits as closed curves. In the plane this is a strong restriction, and it leads to the Poincaré–Bendixson theorem of a later section. Nullclines help to draw portraits: on the \(x_1\)-nullcline the flow is vertical, on the \(x_2\)-nullcline horizontal, and equilibria lie where they cross.
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