12.3 Ruling out periodic orbits
Theorem 12.3.1 (Bendixson–Dulac criterion). Let \(D\) be a simply connected region of the plane and \(B\) a \(C^1\) function on \(D\). If \(\operatorname {div}(Bf) = \partial _1(Bf_1) + \partial _2(Bf_2)\) has one sign on \(D\) and is not identically zero on any open subset, then \(D\) contains no periodic orbit of \(x' = f(x)\).
Proof. Suppose \(\Gamma \subset D\) is a periodic orbit, enclosing the region \(R \subseteq D\) (since \(D\) is simply connected). By Green’s theorem, \[ \iint _R\operatorname {div}(Bf)\,dA = \oint _\Gamma \bigl (Bf_1\,dy - Bf_2\,dx\bigr ) = \int _0^TB\bigl (f_1x_2' - f_2x_1'\bigr )\,dt = \int _0^TB\bigl (f_1f_2 - f_2f_1\bigr )\,dt = 0 , \] using \(dx_1 = f_1\,dt\) and \(dx_2 = f_2\,dt\) along the orbit. But the left side is non-zero, since the integrand has one sign and is not identically zero. This is a contradiction. □
With \(B = 1\) the criterion is Bendixson’s: a planar system whose divergence has one sign has no cycles, which is the planar case of the volume argument of Theorem 11.5.2(iii). The function \(B\) (a Dulac function) extends the reach. For the competition model of Section 10.4, with \(B = 1/(xy)\) on the open positive quadrant, \[ \operatorname {div}(Bf) = \frac {\partial }{\partial x}\Bigl (\frac {3 - x - 2y}{y}\Bigr ) + \frac {\partial }{\partial y}\Bigl (\frac {2 - x - y}{x}\Bigr ) = -\frac 1y - \frac 1x < 0 , \] so there are no periodic orbits, as Figure 14 suggested. The same function works for every two-species Lotka–Volterra competition model.
Index theory
A further necessary condition comes from topology. The index of a closed curve \(C\) not passing through an equilibrium is the number of turns the vector \(f(x)\) makes as \(x\) goes once round \(C\). It is \(+1\) for a small circle around a node, focus or centre, \(-1\) around a saddle, and \(+1\) for a periodic orbit (the vector field is tangent to it). It is also the sum of the indices of the equilibria enclosed. Hence every periodic orbit encloses equilibria whose indices sum to \(1\), at least one of them. A region containing no equilibria, or only saddles, contains no periodic orbit.
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