# Dimension Reduction Near Periodic Orbits of Hybrid Systems: Appendix

### Sam Burden, Shai Revzen and S. Shankar Sastry

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2011-100

September 7, 2011

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-100.pdf

When the Poincare map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in ﬁnite time. This result shows that the long-term behavior of a hybrid model with a large number of degrees-of-freedom may be governed by a low-dimensional smooth dynamical system. The appearance of such simpliﬁed models enables the translation of analytical tools from smooth systems—such as Floquet theory—to the hybrid setting and provides a bridge between the efforts of biologists and engineers studying legged locomotion.

BibTeX citation:

@techreport{Burden:EECS-2011-100, Author = {Burden, Sam and Revzen, Shai and Sastry, S. Shankar}, Title = {Dimension Reduction Near Periodic Orbits of Hybrid Systems: Appendix}, Institution = {EECS Department, University of California, Berkeley}, Year = {2011}, Month = {Sep}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-100.html}, Number = {UCB/EECS-2011-100}, Abstract = {When the Poincare map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in ﬁnite time. This result shows that the long-term behavior of a hybrid model with a large number of degrees-of-freedom may be governed by a low-dimensional smooth dynamical system. The appearance of such simpliﬁed models enables the translation of analytical tools from smooth systems—such as Floquet theory—to the hybrid setting and provides a bridge between the efforts of biologists and engineers studying legged locomotion.} }

EndNote citation:

%0 Report %A Burden, Sam %A Revzen, Shai %A Sastry, S. Shankar %T Dimension Reduction Near Periodic Orbits of Hybrid Systems: Appendix %I EECS Department, University of California, Berkeley %D 2011 %8 September 7 %@ UCB/EECS-2011-100 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-100.html %F Burden:EECS-2011-100