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://www2.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://www2.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://www2.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-100.html
%F Burden:EECS-2011-100