On Poincare-bendixson Theorem And Closed Orbits Of Nonlinear Dynamical Systems

In a certain sense, closed orbits are the only types of orbits that we can ever hopernto understand completely throughout their evolution from the distant past (i.e. asrnt ! −1) to the distant future (i.e., as t ! 1) since the entire course of theirrnevolution is determined by knowledge over a finite time interval, i.e., the period. Likernequilibrium points that are asymptotically stable, periodic solutions may also attractrnother solutions. Determining the long time behavior of closed orbits is much morerndifficult while in this project paper we do have a tool that resembles the linearizationrntechnique called the Poincar´e map. Furthermore, by applying the Poincar´e-Bendixsonrntheorem the limiting behaviors of a planar flow is determined.

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