Border-Collision bifurcations in One-Dimensional Discontinuous Maps

We present a classification of border-collision bifurcations in one-dimensional discontinuous maps depending on the parameters of the piecewise linear approximation in the neighborhood of the point of discontinuity. For each range of parameter values we derive the condition of existence and stability of various periodic orbits and of chaos. This knowledge will help in understanding the bifurcation phenomena in a large number of practical systems which can be modeled by discontinuous maps in discrete domain.

[1]  James A. Yorke,et al.  BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS , 1995 .

[2]  James A. Yorke,et al.  Border-collision bifurcations in the buck converter , 1998 .

[3]  Michael Peter Kennedy,et al.  Nonsmooth bifurcations in a piecewise linear model of the Colpitts Oscillator , 2000 .

[4]  I. Dobson,et al.  Nonlinear dynamics and switching time bifurcations of a thyristor controlled reactor circuit , 1996 .

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

[6]  Ott,et al.  Border-collision bifurcations: An explanation for observed bifurcation phenomena. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  I. Dobson.  Stability of ideal thyristor and diode switching circuits , 1995 .

[8]  James A. Yorke,et al.  Anomalous bifurcations in DC-DC converters: borderline collisions in piecewise smooth maps , 1997, PESC97. Record 28th Annual IEEE Power Electronics Specialists Conference. Formerly Power Conditioning Specialists Conference 1970-71. Power Processing and Electronic Specialists Conference 1972.

[9]  Chris Budd,et al.  Chattering and related behaviour in impact oscillators , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[10]  Celso Grebogi,et al.  Universal behavior of impact oscillators near grazing incidence , 1995 .

[11]  J. Yorke,et al.  Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits , 2000 .

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

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

[14]  George C. Verghese,et al.  Nonlinear Phenomena in Power Electronics , 2001 .

[15]  Chris Budd.  Grazing in Impact Oscillators , 1995 .

[16]  Ian Dobson,et al.  Switching time bifurcations in a thyristor controlled reactor , 1996 .