Figure 1.
The orbit of the periodic orbit for the system given by Theorem 5.2
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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 |
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, doi: 10.3934/mcrf.2019028
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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.
|
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. |
| [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.
|
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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