Notes on the Poincaré–bendixson Theorem

Our goal in these notes is to understand the long-time behavior of solutions to ODEs. For this it will be very useful to introduce the notion of ω-limit sets. A remarkable result the Poincaré–Bendixson theorem is that for planar ODEs, one can have a rather good understanding of ω-limit sets. I have been benefited a lot from the textbook Differential equations, dynamical systems and an introduction to chaos by Hirsch– Smale–Devaney while preparing for these notes. As alwyas, this is a preliminary version, if you have any comments or corrections, even very minor ones, please let me know. Consider the following ODE, i.e., u′(t) = F (u(t)), (0.1) where F : R → R is C. In particular, the Picard–Lindelöf theorem applies. Therefore, we know that given any initial data, there exists a unique maximal solution.