Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map

Abstract A two-dimensional piecewise linear continuous model is analyzed. It reflects the dynamics occurring in a circuit proposed as chaos generator, in a simplified case. The parameter space is investigated in order to classify completely regions of existence of stable cycles, and regions associated with chaotic behaviors. The border collision bifurcation curves are analytically detected, as well as the degenerate flip bifurcations of k-cycles and the homoclinic bifurcations occurring in cyclic chaotic regions leading to chaos in one-piece.

[1]  Pascal Chargé,et al.  Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit. , 2011, Chaos.

[2]  Celso Grebogi,et al.  Border collision bifurcations in two-dimensional piecewise smooth maps , 1998, chao-dyn/9808016.

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

[4]  Hiroshi Kawakami,et al.  Experimental Realization of Controlling Chaos in the Periodically Switched Nonlinear Circuit , 2004, Int. J. Bifurc. Chaos.

[5]  Ahmed S. Elwakil,et al.  An equation for Generating Chaos and its monolithic Implementation , 2002, Int. J. Bifurc. Chaos.

[6]  Mitrajit Dutta,et al.  Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems , 1999 .

[7]  Laura Gardini,et al.  Degenerate bifurcations and Border Collisions in Piecewise Smooth 1D and 2D Maps , 2010, Int. J. Bifurc. Chaos.

[8]  S. Kolyada,et al.  On dynamics of triangular maps of the square , 1992, Ergodic Theory and Dynamical Systems.

[9]  Soumitro Banerjee,et al.  Dangerous bifurcation at border collision: when does it occur? , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[11]  Ahmed S. Elwakil,et al.  New chaos generators , 1997 .

[12]  L. Gardini,et al.  A Goodwin-Type Model with a Piecewise Linear Investment Function , 2006 .

[13]  Soumitro Banerjee,et al.  Robust Chaos , 1998, chao-dyn/9803001.

[14]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .

[15]  James D. Meiss,et al.  Neimark-Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps , 2008, SIAM J. Appl. Dyn. Syst..

[16]  T. Kousaka,et al.  Analysis of border-collision bifurcation in a simple circuit , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[17]  Leon O. Chua,et al.  Cycles of Chaotic Intervals in a Time-delayed Chua's Circuit , 1993, Chua's Circuit.

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

[19]  Jaroslav Smítal,et al.  ω-Limit sets for triangular mappings , 2001 .

[20]  Piotr Kowalczyk,et al.  Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps , 2005 .

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

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

[23]  Ekaterina Pavlovskaia,et al.  Experimental study of impact oscillator with one-sided elastic constraint , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[24]  Ekaterina Pavlovskaia,et al.  Invisible grazings and dangerous bifurcations in impacting systems: the problem of narrow-band chaos. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  L. Chua,et al.  A universal circuit for studying and generating chaos. I. Routes to chaos , 1993 .

[26]  Laura Gardini,et al.  Cournot duopoly when the competitors operate multiple production plants , 2009 .

[27]  Ángel Rodríguez-Vázquez,et al.  Integrated chaos generators , 2002 .

[28]  Ángel Rodríguez-Vázquez,et al.  Mixed-signal map-configurable integrated chaos generator for chaotic communications , 2001 .

[29]  Laura Gardini,et al.  Center bifurcation for Two-Dimensional Border-Collision Normal Form , 2008, Int. J. Bifurc. Chaos.