Dimension Reduction Near Periodic Orbits of Hybrid Systems

Details Dimension Reduction Near Periodic Orbits of Hybrid Systems Dimension Reduction Near Periodic Orbits of Hybrid Systems

2011-09-08

arxiv.org/abs/1109.1780v1

Mathematics

Dynamical Systems

Full version of conference paper appearing in IEEE CDC/ECC 2011

Scientific paper

When the Poincar\'{e} 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.

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