Bifurcation of periodic orbits in a class of planar Filippov systems

Abstract In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.

[1]  Mario di Bernardo,et al.  Sliding bifurcations: a Novel Mechanism for the Sudden Onset of Chaos in dry Friction oscillators , 2003, Int. J. Bifurc. Chaos.

[2]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[3]  Zhengdong Du,et al.  Asymmetric type II periodic motions for nonlinear impact oscillators , 2008 .

[4]  M. I. Feigin,et al.  Doubling of the oscillation period with C-bifurcations in piecewise-continuous systems: PMM vol. 34, n≗5, 1970, pp. 861–869 , 1970 .

[5]  Karl Henrik Johansson,et al.  Fast switches in relay feedback systems , 1999, Autom..

[6]  Wolf-Jürgen Beyn,et al.  Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin , 2006, J. Nonlinear Sci..

[7]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[8]  Tassilo Küpper,et al.  Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems , 2005 .

[9]  H. Dankowicz,et al.  Control of near-grazing dynamics in impact oscillators , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[11]  van Dh Dick Campen,et al.  Bifurcation phenomena in non-smooth dynamical systems , 2006 .

[12]  Emilio Freire.  The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems , 2005, SIAM J. Appl. Math..

[13]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[14]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[15]  Jianhua Xie,et al.  Bifurcations and chaos in a system with impacts , 2001 .

[16]  F. Vasca,et al.  Bifurcations in piecewise-smooth feedback systems , 2002 .

[17]  Karl Johan Åström,et al.  Oscillations in Systems with Relay Feedback , 1993 .

[18]  Carmen Chicone,et al.  Bifurcations of nonlinear oscillations and frequency entrainment near resonance , 1992 .

[19]  Tong-heng Lee,et al.  Local Stability of Limit Cycles for MIMO Relay Feedback Systems , 2003, 2003 4th International Conference on Control and Automation Proceedings.

[20]  M. Kunze.  Non-Smooth Dynamical Systems , 2000 .

[21]  Zhengdong Du,et al.  Melnikov method for homoclinic bifurcation in nonlinear impact oscillators , 2005 .

[22]  Stephen John Hogan,et al.  Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems , 1999 .

[23]  H. Dankowicz,et al.  On the origin and bifurcations of stick-slip oscillations , 2000 .

[24]  Peter Kukučka,et al.  Melnikov method for discontinuous planar systems , 2007 .

[25]  G. Verghese,et al.  Nonlinear phenomena in power electronics : attractors, bifurcations, chaos, and nonlinear control , 2001 .

[26]  Tryphon T. Georgiou,et al.  Dynamics of relay relaxation oscillators , 2001, IEEE Trans. Autom. Control..

[27]  Zhengdong Du,et al.  Type I periodic motions for nonlinear impact oscillators , 2007 .

[28]  Mario di Bernardo,et al.  Grazing, skipping and sliding: Analysis of the non-smooth dynamics of the DC/DC buck converter , 1998 .

[29]  Karl Henrik Johansson,et al.  Stability of limit cycles with chattering in relay feedback systems , 2003, 2003 European Control Conference (ECC).

[30]  K.J. Astrom,et al.  Recent advances in relay feedback methods-a survey , 1995, 1995 IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century.