Planar dynamics and Poincaré–Bendixson

Let f be continuously differentiable on an open subset U of the plane. Suppose a forward orbit remains in a compact subset of U, and its omega-limit set contains no equilibrium. The Poincaré–Bendixson theorem says that omega-limit set is a single periodic orbit. Compact containment ensures forward existence and a nonempty compact limit set. The no-equilibrium condition excludes the stationary and connecting-orbit alternatives. This is a statement about continuous-time autonomous planar flows, not arbitrary maps or projected higher-dimensional motion.

A compact positively invariant region with no equilibria gives a practical existence argument: pick any initial state there, obtain its nonempty limit set inside the region, and apply the theorem. The conclusion alone need not imply that the periodic orbit is isolated or attracting from an open neighborhood. A center has a whole family of periodic orbits. A limit cycle is an isolated periodic orbit, and its stability requires additional information such as transverse contraction or a return-map derivative.

Use polar equations r dot = r(4−r squared) and theta dot = one. They arise from the smooth Cartesian polynomial field x dot = (4−r squared)x−y and y dot = x+(4−r squared)y. At radius one the radial velocity is positive; at radius three it is negative. The closed annulus between them is therefore positively invariant. Angular velocity never vanishes there, so no point in the annulus is an equilibrium. The origin is an equilibrium of the full field but lies outside the trapping region.

Poincaré–Bendixson now guarantees a periodic orbit in the annulus. The explicit radial equation identifies it more precisely: radius two is stationary, smaller positive radii increase, and larger radii decrease. Thus that circle is isolated and attracting for all nonzero initial radii, with period two pi. This additional radial argument establishes uniqueness among periodic radii and stability; these properties should not be attributed to the existence theorem alone. The angle keeps rotating while the radial error decays.

Bendixson's criterion gives a complementary obstruction. If a continuously differentiable planar field has divergence strictly positive everywhere, or strictly negative everywhere, on a simply connected open region D, no nonconstant periodic orbit can lie wholly in D. A hypothetical periodic orbit would bound an interior in D; the field is tangent along its boundary, so its net outward flux is zero, contradicting the area integral of the strictly signed divergence. This argument is a direct use of the planar divergence theorem.

Simply connectedness ensures that the bounded interior belongs to the domain where the divergence sign is controlled. An annulus can surround a missing region, so that hypothesis cannot be dropped without replacement. A sign-changing or identically zero divergence also makes this simple criterion inconclusive rather than proving that cycles exist. For periodically forced planar systems, adding the forcing phase gives a three-dimensional autonomous description, so neither planar obstruction can be applied to the two displayed coordinates as if the clock had disappeared.