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

A Poincaré-Bendixson theorem for hybrid systems

William Clark 1,, , Anthony Bloch 1, and  Leonardo Colombo 2,

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  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.

Keywords: Poincaré-Bendixson theorem, hybrid systems, hybrid flows, stability of periodic orbits, Poincaré map.
Mathematics Subject Classification: Primary: 34A38, 34C25, 70K05; Secondary: 34D20, 70K42.
Citation: William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & 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.  Google Scholar

[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.  Google Scholar

[3]

M. Coleman, A Stability Study of a Three-Dimensional Passive-dynamic Model of Human Gait, Ph.D. dissertation, Cornell University, 1998.  Google Scholar

[4]

S. Collins, A. Ruina, R. Tedrake and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, 307 (5712), 1082-1085.   Google Scholar

[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.  Google Scholar

[6]

R. Goebel, R. Sanfelice and A. Teel, Hybrid Dynamical Systems, Princeton University Press. 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] M. Hirsch, S. Smale and R. Devaney., Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

E. A. Lacomba and W. M. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A, 23 (1990), 2801-2813.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

A. Matveev and A. Savkin, Qualitative Theory of Hybrid Dynamical Systems, Birkhauser Boston, 2000. Google Scholar

[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.   Google Scholar

[18]

L. Perko, Differential Equations and Dynamical Systems, Texts in applied mathematics. Springer-Verlag, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[19]

H. Poincaré, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl., 2 (1886), 151-217.   Google Scholar

[20]

W. Rudin, Principles of Mathematical Analysis, International series in pure and applied mathematics. McGraw-Hill, 1976.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015.   Google Scholar

show all references

References:
[1]

I. Bendixson, Sur les courbes definies par des equations differentielles, Acta Mathematica, 24 (1901), 1-88.  doi: 10.1007/BF02403068.  Google Scholar

[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.  Google Scholar

[3]

M. Coleman, A Stability Study of a Three-Dimensional Passive-dynamic Model of Human Gait, Ph.D. dissertation, Cornell University, 1998.  Google Scholar

[4]

S. Collins, A. Ruina, R. Tedrake and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, 307 (5712), 1082-1085.   Google Scholar

[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.  Google Scholar

[6]

R. Goebel, R. Sanfelice and A. Teel, Hybrid Dynamical Systems, Princeton University Press. 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] M. Hirsch, S. Smale and R. Devaney., Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

E. A. Lacomba and W. M. Tulczyjew, Geometric formulation of mechanical systems with one-sided constraints, J. Phys. A, 23 (1990), 2801-2813.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

A. Matveev and A. Savkin, Qualitative Theory of Hybrid Dynamical Systems, Birkhauser Boston, 2000. Google Scholar

[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.   Google Scholar

[18]

L. Perko, Differential Equations and Dynamical Systems, Texts in applied mathematics. Springer-Verlag, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[19]

H. Poincaré, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl., 2 (1886), 151-217.   Google Scholar

[20]

W. Rudin, Principles of Mathematical Analysis, International series in pure and applied mathematics. McGraw-Hill, 1976.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015.   Google Scholar
Figure 1.  The orbit of the periodic orbit for the system given by Theorem 5.2
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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 $x$
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Figure 3.  1000 cycles of the flow from §5.1.1
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Figure 4.  Displaying the locations of the jumps after performing 1000 iterations of the system in §5.1.2
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Figure 5.  The rimless wheel
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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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