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# A Poincaré-Bendixson theorem for hybrid systems

• * Corresponding author: William Clark

W. Clark was supported by NSF grant DMS-1613819. A. Bloch was supported by NSF grant DMS-1613819 and AFOSR grant FA 9550-18-0028. L. Colombo was partially supported by Ministerio de Economia, Industria y Competitividad (MINEICO, Spain) under grant MTM2016-76702-P and "Severo Ochoa Programme for Centres of Excellence" in R & D (SEV-2015-0554)

• 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.

Mathematics Subject Classification: Primary: 34A38, 34C25, 70K05; Secondary: 34D20, 70K42.

 Citation: • • Figure 1.  The orbit of the periodic orbit for the system given by Theorem 5.2

Figure 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 $x$

Figure 3.  1000 cycles of the flow from §5.1.1

Figure 4.  Displaying the locations of the jumps after performing 1000 iterations of the system in §5.1.2

Figure 5.  The rimless wheel

Figure 6.  Left: The lighter region indicates values of $\alpha$ and $\delta$ where there exists a limit cycle as predicted by equation (34). Right: The lower region is the domain of attraction for the limit cycle, whose existence is guaranteed by equation (34)

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