Generalized Hopf Bifurcation for Planar Filippov Systems
Continuous at the Origin |
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Authors: | Y Zou T Kupper W-J Beyn |
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Institution: | (1) Department of Mathematics, Jilin University, Changchun 130023, P. R. China;(2) Mathematisches Institut, der Universitat zu Koln, Weyertal 86-90, D-50931 Koln, Germany;(3) Fakultat fur Mathematik, Universitat Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany |
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Abstract: | In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system
of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation
is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry
over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In
fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures
of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical
method to investigate the existence of periodic orbits that are
obtained by searching for the fixed points of return maps. |
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Keywords: | Filippov system Hopf bifurcation |
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