On Poincar´e-Bendixson Theorem and Closed Orbits of Nonlinear Dynamical Systems
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Date
2012-02
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Addis Ababa University
Abstract
In a certain sense, closed orbits are the only types of orbits that we can ever hope
to understand completely throughout their evolution from the distant past (i.e. as
t ! −1) to the distant future (i.e., as t ! 1) since the entire course of their
evolution is determined by knowledge over a finite time interval, i.e., the period. Like
equilibrium points that are asymptotically stable, periodic solutions may also attract
other solutions. Determining the long time behavior of closed orbits is much more
difficult while in this project paper we do have a tool that resembles the linearization
technique called the Poincar´e map. Furthermore, by applying the Poincar´e-Bendixson
theorem the limiting behaviors of a planar flow is determined.
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On Poincar´e-Bendixson Theorem, and Closed Orbits of Nonlinear