Dynamics of a piecewise linear map with a gap

In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13, 3341–3351) and Banerjee et al. (Banerjee et al. 2004 IEEE Trans. Circ. Syst. II 51, 649–654) and explain their numerical simulations.

[1]  Michael Schanz,et al.  Period-Doubling Scenario without flip bifurcations in a One-Dimensional Map , 2005, Int. J. Bifurc. Chaos.

[2]  V. N. Belykh,et al.  Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models , 2000 .

[3]  Amit Gupta,et al.  Dynamical effects of missed switching in current-mode controlled DC-DC converters , 2004, IEEE Transactions on Circuits and Systems II: Express Briefs.

[4]  J Wang,et al.  Characteristics of a piecewise smooth area-preserving map. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Mario di Bernardo,et al.  The importance of choosing attractors for optimizing chaotic communications , 2004, IEEE Transactions on Circuits and Systems II: Express Briefs.

[6]  T. Lofaro,et al.  Period-adding bifurcations in a one parameter family of interval maps , 1996 .

[7]  Michael Schanz,et al.  Border-collision period-doubling scenario. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  J. Keener Chaotic behavior in piecewise continuous difference equations , 1980 .

[9]  D. He,et al.  Crisis induced by an escape from a fat strange set. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  He,et al.  Dynamic interaction between discontinuity and noninvertibility: An analytical study. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Y. Lai,et al.  Basins of attraction in piecewise smooth Hamiltonian systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Soumitro Banerjee,et al.  Border-Collision bifurcations in One-Dimensional Discontinuous Maps , 2003, Int. J. Bifurc. Chaos.

[13]  Alan R. Champneys,et al.  Bifurcations in piecewise-smooth dynamical systems: Theory and Applications , 2007 .

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

[15]  Gábor Stépán,et al.  Dynamics of Piecewise Linear Discontinuous Maps , 2004, Int. J. Bifurc. Chaos.

[16]  Da-Ren He,et al.  A multiple devil's staircase in a discontinuous map , 1997 .

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

[18]  Christiansen,et al.  Phase diagram of a modulated relaxation oscillator with a finite resetting time. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[19]  D. He,et al.  Multiple devil's staircase and type-V intermittency , 1998 .

[20]  Erik Mosekilde,et al.  Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems: Applications to Power Converters, Relay and Pulse-Width Modulated Control Systems, and Human Decision-Making Behavior , 2003 .