Period Adding in Piecewise Linear Maps with Two Discontinuities

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

[1]  C. Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[2]  L. Gardini,et al.  Cournot Duopoly with Kinked Demand According to Palander and Wald , 2002 .

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

[4]  Tönu Puu,et al.  Oligopoly Dynamics : Models and Tools , 2002 .

[5]  O. Feely,et al.  Lowpass sigma–delta modulation: an analysis by means of the critical lines tool , 2001 .

[6]  Iryna Sushko,et al.  Business Cycle Dynamics , 2006 .

[7]  Laura Gardini,et al.  Hicks' trade cycle revisited: cycles and bifurcations , 2003, Math. Comput. Simul..

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

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

[10]  B. Hao,et al.  Elementary Symbolic Dynamics And Chaos In Dissipative Systems , 1989 .

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

[12]  Volodymyr L. Maistrenko,et al.  On period-adding sequences of attracting cycles in piecewise linear maps , 1998 .

[13]  Michael Schanz,et al.  Calculation of bifurcation Curves by Map Replacement , 2010, Int. J. Bifurc. Chaos.

[14]  Michael Schanz,et al.  On multi-parametric bifurcations in a scalar piecewise-linear map , 2006 .

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

[16]  Laura Gardini,et al.  Growing through chaotic intervals , 2008, J. Econ. Theory.

[17]  Leon O. Chua,et al.  BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT , 1995 .

[18]  Richard H. Day,et al.  Complex economic dynamics , 1994 .

[19]  Cars H. Hommes,et al.  “Period three to period two” bifurcation for piecewise linear models , 1991 .

[20]  Laura Gardini,et al.  The Hicksian floor-roof model for two regions linked by interregional trade , 2003 .

[21]  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 .

[22]  Cars H. Hommes,et al.  Cycles and chaos in a socialist economy , 1995 .

[23]  T. Schelling.  Micromotives and Macrobehavior , 1978 .

[24]  T. Schelling.  Hockey Helmets, Concealed Weapons, and Daylight Saving , 1973 .

[25]  Laura Gardini,et al.  Inertia in binary choices: Continuity breaking and big-bang bifurcation points , 2011 .

[26]  Orla Feely,et al.  Nonlinear Dynamics of Bandpass Sigma-Delta Modulation - an Investigation by Means of the Critical Lines Tool , 2000, Int. J. Bifurc. Chaos.

[27]  L. Gardini,et al.  The Hicksian Model with Investment Floor and Income Ceiling , 2006 .

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

[29]  Erik Mosekilde,et al.  Quasiperiodicity and torus breakdown in a power electronic dc/dc converter , 2007, Math. Comput. Simul..

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

[31]  Orla Feely,et al.  Nonlinear dynamics of bandpass sigma-delta modulation , 1996, 1996 8th European Signal Processing Conference (EUSIPCO 1996).

[32]  Debra Hevenstone.  Employment Intermediaries: A Model of Firm Incentives , 2008 .

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

[34]  L. Gardini,et al.  Bifurcation Curves in Discontinuous Maps , 2009 .

[35]  Laura Gardini,et al.  BISTABILITY AND BORDER-COLLISION BIFURCATIONS FOR A FAMILY OF UNIMODAL PIECEWISE SMOOTH MAPS , 2005 .

[36]  Laura Gardini,et al.  Tongues of periodicity in a family of two-dimensional discontinuous maps of real Möbius type , 2003 .

[37]  Somnath Maity,et al.  Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. , 2006, Chaos.

[38]  Laura Gardini,et al.  Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities , 2009 .

[39]  T. Puu.  The Hicksian trade cycle with floor and ceiling dependent on capital stock , 2007 .

[40]  Laura Gardini,et al.  Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves , 2006 .

[41]  Tönu Puu,et al.  Business Cycle Dynamics : Models and Tools , 2006 .

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

[43]  Cars H. Hommes,et al.  A reconsideration of Hicks' non-linear trade cycle model , 1995 .

[44]  L. Gardini,et al.  A Hicksian multiplier-accelerator model with floor determined by capital stock , 2005 .

[45]  Michael Schanz,et al.  Multi-parametric bifurcations in a piecewise–linear discontinuous map , 2006 .

[46]  Mario di Bernardo,et al.  C-bifurcations and period-adding in one-dimensional piecewise-smooth maps , 2003 .

[47]  Michael Schanz,et al.  Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach , 2010, Int. J. Bifurc. Chaos.

[48]  Laura Gardini,et al.  Border Collision bifurcations in 1D PWL Map with One Discontinuity and Negative Jump: Use of the First Return Map , 2010, Int. J. Bifurc. Chaos.

[49]  Ugo Merlone,et al.  Global Dynamics in Binary Choice Models with Social Influence , 2009 .

[50]  R. Day.  Irregular Growth Cycles , 2016 .

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