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Media type:
E-Article
Title:
Generalized Hopf Bifurcation for Non-Smooth Planar Systems
Contributor:
Küpper, Tassilo;
Moritz, Susanne
Published:
The Royal Society, 2001
Published in:
Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 359 (2001) 1789, Seite 2483-2496
Language:
English
ISSN:
1364-503X
Origination:
Footnote:
Description:
Hopf bifurcation for smooth systems is characterized by a crossing of a pair of complex conjugate eigenvalues of the linearized problem through the imaginary axis. Since this approach is not at hand for non-smooth systems, we use the geometrical characterization given by the change from an unstable to a stable focus through a centre for a basic (piecewise) linear system. In that way we find two mechanisms for the destabilizing of the basic stationary solution and for the generation of bifurcating periodic orbits: a generation switch of the stability properties or the influence of the unstable subsystem measured by the time of duration spent in the subsystem. The switch between stable and unstable subsystems seems to be a general source of destabilization observed in several mechanical systems. We expect that the features analysed for planar systems will help us to understand higher-dimensional systems as well.