Poincare–Bendixson theorem

Any text on nonlinear systems can hardly afford, not to refer to the celebrated result called ‘Poincare–Bendixson Theorem’. It was first established by the French mathematician Henri Poincare, in 1880 and later on, simplified by the Swiss mathematician Ivan Bendixson in 1901. It is known to be an indispensably vital result that asserts the existence of a periodic solution for a general class of non-linear systems. The theorem has been stated earlier without a formal proof, in connection with periodic orbits, which, of course, focus its primary concern. The ideas in its proof are relatively difficult and so, the proof is not set forth on the first exposure to this theorem. Perhaps, it will now be fairly comfortable to have glimpses of ways and ideas leading to the proof of this theorem. The proofs are achieved apparently in different ways but they do depend on concepts and results treated in the early part of this book. Poincare-Bendixson theorem (PB) has, on the face of it, different versions, as shown below in different sections.

Revisiting concepts treated earlier

We look upon the PB theorem as a basic wherewithal even for understanding planar dynamical systems, which we have already met. For this, we need to bank upon some properties of the limiting behaviour of orbits at the level of abstract topological dynamics, followed by an analysis of the flow near non-equilibrium points of a dynamical system.