# American Institute of Mathematical Sciences

March  2020, 10(1): 27-45. doi: 10.3934/mcrf.2019028

## A Poincaré-Bendixson theorem for hybrid systems

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, USA 2 Instituto de Ciencias Mathemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049, Madrid, Spain

* Corresponding author: William Clark

Received  April 2018 Published  March 2020 Early access  April 2019

Fund Project: 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.

Citation: William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control and Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028
##### References:
 [1] I. Bendixson, Sur les courbes definies par des equations differentielles, Acta Mathematica, 24 (1901), 1-88.  doi: 10.1007/BF02403068. [2] A. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics. Springer New York, 2003. doi: 10.1007/b97376. [3] M. Coleman, A Stability Study of a Three-Dimensional Passive-dynamic Model of Human Gait, Ph.D. dissertation, Cornell University, 1998. [4] S. Collins, A. Ruina, R. Tedrake and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, 307 (5712), 1082-1085. [5] J. Cortés and A. M. Vinogradov, Hamiltonian theory of constrained impulsive motion,, J. Math. Phys., 47 (2006), 042905, 30pp. doi: 10.1063/1.2192974. [6] R. Goebel, R. Sanfelice and A. Teel, Hybrid Dynamical Systems, Princeton University Press. 2012. [7] J. W. Grizzle, G. Abba and F. Plestan, Asymptotically stable walking for biped robots analysis via systems with impulse effects, IEEE Transactions on Automatic Control, 46 (2001), 51-64.  doi: 10.1109/9.898695. [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [9] M. Hirsch, S. Smale and R. Devaney., Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004. [10] P. Holmes, R. Full, D. Koditschek and J. Guckenheimer, Dynamics of legged locomotion: Models, analyses, and challenges, SIAM Review, 48 (2006), 207-304.  doi: 10.1137/S0036144504445133. [11] A. Ibort, M. de León, E. Lacomba, D. de Diego and P. Pitanga, Mechanical systems subjected to impulsive constraints,, J. Phys. A, 30 (1997), 5835-5854.  doi: 10.1088/0305-4470/30/16/024. [12] E. A. Lacomba and W. M. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A, 23 (1990), 2801-2813. [13] X. Lou, Y. Li and R. Sanfelice, On robust stability of limit cycles for hybrid systems with multiple jumps, Proceedings of the 5th Analysis and Design of Hybrid Systems, (2015), 199-204. [14] X. Lou, Y. Li and R. Sanfelice, Results on stability and robustness of hybrid limit cycles for a class of hybrid systems, Proceedings IEEE 54th Annual Conference on Decision and Control (CDC), (2015), 2235-2240. [15] X. Lou, Y. Li and R. Sanfelice, Existence of hybrid limit cycles and Zhukovskii stability in hybrid systems, American Control Conference (ACC), (2017), 1187-1192. [16] A. Matveev and A. Savkin, Qualitative Theory of Hybrid Dynamical Systems, Birkhauser Boston, 2000. [17] B. Morris and J. W. Grizzle, A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots, Proceedings IEEE 44th Annual Conference on Decision and Control (CDC), (2005), 4199-4206. [18] L. Perko, Differential Equations and Dynamical Systems, Texts in applied mathematics. Springer-Verlag, 1991. doi: 10.1007/978-1-4684-0392-3. [19] H. Poincaré, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl., 2 (1886), 151-217. [20] W. Rudin, Principles of Mathematical Analysis, International series in pure and applied mathematics. McGraw-Hill, 1976. [21] C. Saglam, A. Teel and K. Byl, Lyapunov-based versus Poincaré map analysis of the rimless wheel, IEEE 53rd Annual Conference on Decision and Control (CDC), (2014), 1514-1520. [22] S. Simic, S. Sastry, K. Johansson and J. Lygeros., Hybrid limit cycles and hybrid Poincaré-Bendixson, 15th IFAC World Congress, 35 (2002), 197-202. [23] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015.

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##### References:
 [1] I. Bendixson, Sur les courbes definies par des equations differentielles, Acta Mathematica, 24 (1901), 1-88.  doi: 10.1007/BF02403068. [2] A. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics. Springer New York, 2003. doi: 10.1007/b97376. [3] M. Coleman, A Stability Study of a Three-Dimensional Passive-dynamic Model of Human Gait, Ph.D. dissertation, Cornell University, 1998. [4] S. Collins, A. Ruina, R. Tedrake and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, 307 (5712), 1082-1085. [5] J. Cortés and A. M. Vinogradov, Hamiltonian theory of constrained impulsive motion,, J. Math. Phys., 47 (2006), 042905, 30pp. doi: 10.1063/1.2192974. [6] R. Goebel, R. Sanfelice and A. Teel, Hybrid Dynamical Systems, Princeton University Press. 2012. [7] J. W. Grizzle, G. Abba and F. Plestan, Asymptotically stable walking for biped robots analysis via systems with impulse effects, IEEE Transactions on Automatic Control, 46 (2001), 51-64.  doi: 10.1109/9.898695. [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [9] M. Hirsch, S. Smale and R. Devaney., Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004. [10] P. Holmes, R. Full, D. Koditschek and J. Guckenheimer, Dynamics of legged locomotion: Models, analyses, and challenges, SIAM Review, 48 (2006), 207-304.  doi: 10.1137/S0036144504445133. [11] A. Ibort, M. de León, E. Lacomba, D. de Diego and P. Pitanga, Mechanical systems subjected to impulsive constraints,, J. Phys. A, 30 (1997), 5835-5854.  doi: 10.1088/0305-4470/30/16/024. [12] E. A. Lacomba and W. M. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A, 23 (1990), 2801-2813. [13] X. Lou, Y. Li and R. Sanfelice, On robust stability of limit cycles for hybrid systems with multiple jumps, Proceedings of the 5th Analysis and Design of Hybrid Systems, (2015), 199-204. [14] X. Lou, Y. Li and R. Sanfelice, Results on stability and robustness of hybrid limit cycles for a class of hybrid systems, Proceedings IEEE 54th Annual Conference on Decision and Control (CDC), (2015), 2235-2240. [15] X. Lou, Y. Li and R. Sanfelice, Existence of hybrid limit cycles and Zhukovskii stability in hybrid systems, American Control Conference (ACC), (2017), 1187-1192. [16] A. Matveev and A. Savkin, Qualitative Theory of Hybrid Dynamical Systems, Birkhauser Boston, 2000. [17] B. Morris and J. W. Grizzle, A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots, Proceedings IEEE 44th Annual Conference on Decision and Control (CDC), (2005), 4199-4206. [18] L. Perko, Differential Equations and Dynamical Systems, Texts in applied mathematics. Springer-Verlag, 1991. doi: 10.1007/978-1-4684-0392-3. [19] H. Poincaré, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl., 2 (1886), 151-217. [20] W. Rudin, Principles of Mathematical Analysis, International series in pure and applied mathematics. McGraw-Hill, 1976. [21] C. Saglam, A. Teel and K. Byl, Lyapunov-based versus Poincaré map analysis of the rimless wheel, IEEE 53rd Annual Conference on Decision and Control (CDC), (2014), 1514-1520. [22] S. Simic, S. Sastry, K. Johansson and J. Lygeros., Hybrid limit cycles and hybrid Poincaré-Bendixson, 15th IFAC World Congress, 35 (2002), 197-202. [23] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015.
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 $\alpha$ 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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