Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps

International Journal of Bifurcation and Chaos

Volumn 20, Issue 7, 2010, Pages 2045-2070

  Sushko, Iryna ^a   Gardini, Laura ^b  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

^b UNIVERSITY OF URBINO   (Italy)

Author keywords

Border collision; degenerate bifurcations; piecewise smooth

Indexed keywords

DYNAMICAL SYSTEMS; PHASE SPACE METHODS;

BORDER COLLISION; BORDER COLLISION BIFURCATION; CODIMENSION; DISCRETE-TIME DYNAMICAL SYSTEMS; GLOBAL PROPERTIES; NEIMARK-SACKER BIFURCATION; PIECEWISE SMOOTH; PIECEWISE SMOOTH MAPS;

BIFURCATION (MATHEMATICS);

EID: 77956618749     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127410026927     Document Type: Article

References (44)

1. Avrutin, V. & Schanz, M. [2006] "Multi-parametric bifurcations in a scalar piecewise-linear map," Nonlinearity 19, 531-552. (Pubitemid 43235035)
2. Avrutin, V., Schanz, M. & Gardini, L. [2009a] "On a special type of border-collision bifurcations occurring at infinity," submitted for publication.
3. Avrutin, V., Schanz, M. & Gardini, L. [2009b] "Calculation of bifurcation curves by map replacement," submitted for publication.
4. Banerjee, S. & Grebogi, C. [1999] "Border-collision bifurcations in two-dimensional piecewise smooth maps," Phys. Rev. E 59, 4052-4061.
5. Banerjee, S., Karthik, M. S., Yuan, G. & Yorke, J. A. [2000a] "Bifurcations in one-dimensional piecewise smooth maps - Theory and applications in switching circuits," IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 47, 389-394.
6. Banerjee, S., Ranjan, P. & Grebogi, C. [2000b] "Bifurcations in two-dimensional piecewise smooth maps - Theory and applications in switching circuits," IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 47, 633-643.
7. De Melo, W. & van Strien, S. [1993] One-Dimensional Dynamics (Springer, NY).
8. di Bernardo, M., Feigin, M. I., Hogan S. J. & Homer, M. E. [1999] "Local analysis of C-bifurcations in ndimensional piecewise smooth dynamical systems," Chaos Solit. Fract. 10, 1881-1908.
9. di Bernardo, M., Budd, C. J., Champneys, A. R. & Kowalczyk, P. [2008] Piecewise-Smooth Dynamical Systems Theory and Applications (Springer-Verlag, London).
10. Feely, O., Fournier-Prunaret, D., Taralova-Roux, I. & Fitzgerald, D. [2000] "Nonlinear dynamics of bandpass sigma-delta modulation. An investigation by means of the critical lines tool," Int. J. Bifurcation and Chaos 10, 303-327.
11. Feigin, M. I. [1970] "Doubling of the oscillation period with C-bifurcations in piecewise-continuous systems," Prikl. Math. Mekh. 34, 861-869 (in Russian).
12. Fournier-Prunaret, D., Feely, O. & Taralova-Roux, I. [2001] "Lowpass sigma-delta modulation: An analysis by means of the critical lines tool," Nonlin. Anal. 47, 5343-5355.
13. Fournier-Prunaret, D., Charge, P. & Gardini, L. [2008] "Bifurcations in a 2D piecewise-smooth map modeling a switching circuit," DIFF-2008 Suzdal, Russia.
14. Gallegati, M., Gardini, L., Puu, T. & Sushko, I. [2003] "Hicks' trade cycle revisited: Cycles and bifurcations," Math. Comput. Simul. 63, 505-527.
15. Gardini, L., Sushko, I. & Naimzada, A. [2008] "Growing through chaotic intervals," J. Econ. Th. 143, 541-557.
16. Gardini, L., Avrutin, V. & Schanz, M. [2009a] "Connection between bifurcations on the Poincar̀e Equator and the dangerous bifurcations," Grazer Mathematische Berichte, to appear.
17. Gardini, L., Tramontana, F., Avrutin, V. & Schanz, M. [2009b] "Border collision bifurcations in 1D PWL map and the Leonov approach," submitted for publication.
18. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, NY).
19. Halse, C., Homer, M. & di Bernardo, M. [2003] "C-bifurcations and period-adding in one-dimensional piecewise-smooth maps," Chaos Solit. Fract. 18, 953-976.
20. Ito, S., Tanaka, S. & Nakada, H. [1979] "On unimodal transformations and chaos II," Tokyo J. Math. 2, 241-259.
21. Kocic, V. L. & Ladas, G. [1993] Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (Kluwer Academic Publishers).
22. Kuznetsov, Y. A. [1998] Elements of Applied Bifurcation Theory (Springer-Verlag, NY).
23. Leonov, N. N. [1959] "Map of the line onto itself," Radiofisika 3, 942-956.
24. Lozi, R. [1978] "Un attracteur étrange du type attracteur de Hénon," J. Phys. Colloque C5 39, supplément au no. 8, 9-10.
25. Maistrenko, Y. L., Maistrenko, V. L. & Chua, L. O. [1993] "Cycles of chaotic intervals in a time-delayed Chua's circuit," Int. J. Bifurcation and Chaos 3, 1557-1572.
26. Maistrenko, Y. L., Maistrenko, V. L., Vikul, S. I. & Chua, L. O. [1995] "Bifurcations of attracting cycles from time-delayed Chua's circuit," Int. J. Bifurcation and Chaos 5, 653-671.
27. Maistrenko, Y., Sushko, I. & Gardini, L. [1998] "About two mechanisms of reunion of chaotic attractors," Chaos Solit. Fract. 9, 1373-1390.
28. Mira, C., Gardini, L., Barugola, A. & Cathala, J. C. [1996] Chaotic Dynamics in Two-Dimensional NonivertibIe Maps, Nonlinear Sciences, Series A (World-Scientific, Singapore).
29. Nusse, H. E. & Yorke, J. A. [1992] "Border-collision bifurcations including period two to period three for piecewise smooth systems," Physica D 57, 39-57.
30. Nusse, H. E. & Yorke, J. A. [1995] "Border-collision bifurcation for piecewise smooth one-dimensional maps," Int. J. Bifurcation and Chaos 5, 189-207.
31. Puu, T. & Sushko, I. [2002] Oligopoly Dynamics, Models and Tools (Springer Verlag, NY).
32. Puu, T. & Sushko, I. [2006] Business Cycle Dynamics, Models and Tools (Springer Verlag, NY).
33. Sharkovsky A. N., Kolyada, S. F., Sivak, A. G. & Fedorenko, V. V. [1997] Dynamics of One-Dimensional Maps (Kluwer Academic, Boston).
34. Simpson, D. J. W. & Meiss, J. D. [2008] "Neimark- Sacker bifurcations in planar, piecewise-smooth, continuous maps," SIAM J. Appl. Dyn. Syst. 7, 795-824.
35. Sushko, I., Maistrenko, Y. & Gardini, L. [1999] "On chaotic attractors at the transition from homeomorphism to endomorphism in a family of twodimensional maps," Grazer Math. Ber. 339, 335-346.
36. Sushko, I., Puu, T. & Gardini, L. [2003] "The Hicksian floor-roofmodel for two regions linked by interregional trade," Chaos Solit. Fract. 18, 593-612.
37. Sushko, I., Agliari, A. & Gardini, L. [2005] "Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps," Discr. Contin. Dyn. Syst. Serie B 5, 881-897.
38. Sushko, I., Agliari, A. & Gardini, L. [2006] "Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves," Dynamic Modelling in Economics and Finance in Honour of Professor Carl Chiarella, eds. Bischi, G. I. & Sushko, I., Chaos Solit. Fract. 29, 756-770.
39. Sushko, I. & Gardini, L. [2008] "Center bifurcation for two-dimensional border-collision normal form," Int. J. Bifurcation and Chaos 18, 1029-1050.
40. Sushko, I. & Gardini, L. [2009] "Center bifurcation of a point on the Poincaré equator," Grazer Mathematische Berichte, to appear.
41. Takens, F. [1987] "Transitions from periodic to strange attractors in constrained equations," in Dynamical Systems and Bifurcation Theory, eds. Camacho, M. I., Pacifico, M. J., Takens, F., Pitman Research Notes in Mathematics Series, Vol.160 (Longman Scientific and Techical), pp. 399-421.
42. Tramontana, F., Gardini, L. & Puu, T. [2008] "Duopoly games with alternative technologies," J. Econ. Dyn. Contr. 33, 250-265.
43. Zhusubaliyev, Z. T. & Mosekilde, E. [2003] Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems (World Scientific, Singapore).
44. Zhusubaliyev, Z. T., Mosekilde, E., Maity, S., Mohanan, S. & Banerjee, S. [2006] "Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation," Chaos 16, 1-11.