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Quantitative approximation properties for the fractional heat equation
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.
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. 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. |
[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. 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. |
[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. |
[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.
![]() |
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. 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. |
[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. 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. |
[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. |
[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.
![]() |






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