The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.
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The orbit of the periodic orbit for the system given by Theorem 5.2
The vector $\delta x = F(y)\delta y + f(x)\delta t$, where the horizontal line is the tangent to $S$ at the point $x$
1000 cycles of the flow from §5.1.1
Displaying the locations of the jumps after performing 1000 iterations of the system in §5.1.2
The rimless wheel
Left: The lighter region indicates values of