Border-collision bifurcations in a generalized piecewise linear-power map

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September 2011, 16(2): 547-567. doi: 10.3934/dcdsb.2011.16.547

Border-collision bifurcations in a generalized piecewise linear-power map

Zhiying Qin ^1, , Jichen Yang ^2, , Soumitro Banerjee ^3, and Guirong Jiang ^4,

1.	School of Mechanical and Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang 050018, China
2.	Department of Dynamics and Control, Beihang University, Beijing 100191, China
3.	Indian Institute of Science Education & Research, Kolkata Mohanpur-741252, India
4.	Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004

Received February 2010 Revised October 2010 Published June 2011

Abstract
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In this paper a class of generalized piecewise smooth maps is studied, which is linear at one side and nonlinear with power dependence at the other side. According to the value of the power in the term $x^z$, the bifurcations occurring in this map are classified into five types: $z>1$, $z=1$, $0<z<1$, $z=0$, and $z<0$. We derive the occurrence conditions of border collision bifurcation and smooth fold and flip bifurcation, especially the codimension-2 bifurcation points describing the interaction between border collision bifurcation and smooth bifurcation. The general results are then applied to the specific cases of the power $z$, and different bifurcation scenarios are shown for individual cases, from which the period-adding scenario is found to be general for any power.

Keywords: nonsmooth maps., Bifurcation theory, border collision bifurcation.

Mathematics Subject Classification: Primary: 37E05, 37E15, 37G3.

Citation: Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547

References:

[1]	S. Banerjee and G. C. Verghese, "Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations," 1st edition, IEEE, 2001.
[2]	M. di Bernardo, C. J. Budd and A. R Champneys, Corner collision implies border-collision bifurcation, Physica D, 154 (2001), 171-194. doi: 10.1016/S0167-2789(01)00250-0.
[3]	P. T. Piiroinen and C. J. Budd, Corner bifurcations in non-smoothly forced impact oscillators, Physica D, 220 (2006), 127-145. doi: 10.1016/j.physd.2006.07.001.
[4]	A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.
[5]	A. B. Nordmark, Universal limit mapping in grazing bifurcations, Physical Review E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.
[6]	H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations, Physica D, 136 (2000), 280-302. doi: 10.1016/S0167-2789(99)00161-X.
[7]	M. di Bernardo, P. Kowalczyk and A. B. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Physica D, 170 (2002), 170-175.
[8]	M. di Bernardo, C. Budd and A. Champneys, Grazing, skipping and sliding: analysis of the non-smooth dynamics of the dc-dc buck converter, Nonlinearity, 11 (1998), 858-890.
[9]	F. Angulo and M. di Bernardo, Feedback control of limit cycles: A switching control strategy based on nonsmooth bifurcation theory, IEEE Transactions on Circuits and Systems-I, 52 (2005), 366-378. doi: 10.1109/TCSI.2004.841595.
[10]	G. M. Maggio and M. di Bernardo and M. P. Kennedy, Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator, IEEE Transactions on Circuits and Systems-I, 47 (2000), 1160-1177. doi: 10.1109/81.873871.
[11]	P. Jain and S. Banerjee, Border collision bifurcations in one-dimensional discontinuous maps, International Journal of Bifurcation and Chaos, 13 (2003), 3341-3352. doi: 10.1142/S0218127403008533.
[12]	A. Kumar, S. Banerjee and D. P. Lathrop, Dynamics of a piecewise smooth map with sigularity, Physics Letters A, 337 (2005), 87-92. doi: 10.1016/j.physleta.2005.01.046.
[13]	G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps, Discrete and Continuous Dynamical Systems - Series B, 13 (2010), 249-267. doi: 10.3934/dcdsb.2010.13.249.
[14]	H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Physica D, 57 (1992), 39-57. doi: 10.1016/0167-2789(92)90087-4.
[15]	H. E. Nusse and J. A. Yorke, Border-collision bifurcations for piecewise smooth one dimensional maps, International Journal of Bifurcation and Chaos, 5 (1995), 189-207. doi: 10.1142/S0218127495000156.
[16]	M. di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems, Chaos, Solitons and Fractals, 10 (1999), 1881-1908.
[17]	S. Banerjee, M. S. Karthik, G. H. Yuan and J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-I, 47 (2000), 389-394. doi: 10.1109/81.841921.
[18]	S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.
[19]	H. E. Nusse, E. Ott and J. A. Yorke, Border-collision bifurcations: An explanation for observed bifurcation phenomena, Physical Review E, 49 (1994), 1073-1076. doi: 10.1103/PhysRevE.49.1073.
[20]	C. Halse, M. Homer and M. di Bernardo, C-bifurcation and period-adding in one-dimensional piecewise smooth maps, Chaos, Solitons and Fractals, 18 (2003), 953-976. doi: 10.1016/S0960-0779(03)00066-3.
[21]	P. S. Dutta and S. Banerjee, Period increment cascades in a discontinuous map with square-root singularity, Discrete and Continuous Dynamical Systems - Series B, 14 (2010), 961-976. doi: 10.3934/dcdsb.2010.14.961.
[22]	V. Avrutin, M. Schanz and S. Banerjee, Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps, Physical Review E, 75 (2007), 066205. doi: 10.1103/PhysRevE.75.066205.

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References:

[1]	S. Banerjee and G. C. Verghese, "Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations," 1st edition, IEEE, 2001.
[2]	M. di Bernardo, C. J. Budd and A. R Champneys, Corner collision implies border-collision bifurcation, Physica D, 154 (2001), 171-194. doi: 10.1016/S0167-2789(01)00250-0.
[3]	P. T. Piiroinen and C. J. Budd, Corner bifurcations in non-smoothly forced impact oscillators, Physica D, 220 (2006), 127-145. doi: 10.1016/j.physd.2006.07.001.
[4]	A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.
[5]	A. B. Nordmark, Universal limit mapping in grazing bifurcations, Physical Review E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.
[6]	H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations, Physica D, 136 (2000), 280-302. doi: 10.1016/S0167-2789(99)00161-X.
[7]	M. di Bernardo, P. Kowalczyk and A. B. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Physica D, 170 (2002), 170-175.
[8]	M. di Bernardo, C. Budd and A. Champneys, Grazing, skipping and sliding: analysis of the non-smooth dynamics of the dc-dc buck converter, Nonlinearity, 11 (1998), 858-890.
[9]	F. Angulo and M. di Bernardo, Feedback control of limit cycles: A switching control strategy based on nonsmooth bifurcation theory, IEEE Transactions on Circuits and Systems-I, 52 (2005), 366-378. doi: 10.1109/TCSI.2004.841595.
[10]	G. M. Maggio and M. di Bernardo and M. P. Kennedy, Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator, IEEE Transactions on Circuits and Systems-I, 47 (2000), 1160-1177. doi: 10.1109/81.873871.
[11]	P. Jain and S. Banerjee, Border collision bifurcations in one-dimensional discontinuous maps, International Journal of Bifurcation and Chaos, 13 (2003), 3341-3352. doi: 10.1142/S0218127403008533.
[12]	A. Kumar, S. Banerjee and D. P. Lathrop, Dynamics of a piecewise smooth map with sigularity, Physics Letters A, 337 (2005), 87-92. doi: 10.1016/j.physleta.2005.01.046.
[13]	G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps, Discrete and Continuous Dynamical Systems - Series B, 13 (2010), 249-267. doi: 10.3934/dcdsb.2010.13.249.
[14]	H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Physica D, 57 (1992), 39-57. doi: 10.1016/0167-2789(92)90087-4.
[15]	H. E. Nusse and J. A. Yorke, Border-collision bifurcations for piecewise smooth one dimensional maps, International Journal of Bifurcation and Chaos, 5 (1995), 189-207. doi: 10.1142/S0218127495000156.
[16]	M. di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems, Chaos, Solitons and Fractals, 10 (1999), 1881-1908.
[17]	S. Banerjee, M. S. Karthik, G. H. Yuan and J. A. Yorke, Bifurcations in one-dimensional piecewise smooth maps theory and applications in switching circuits, IEEE Transactions on Circuits and Systems-I, 47 (2000), 389-394. doi: 10.1109/81.841921.
[18]	S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps, Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.
[19]	H. E. Nusse, E. Ott and J. A. Yorke, Border-collision bifurcations: An explanation for observed bifurcation phenomena, Physical Review E, 49 (1994), 1073-1076. doi: 10.1103/PhysRevE.49.1073.
[20]	C. Halse, M. Homer and M. di Bernardo, C-bifurcation and period-adding in one-dimensional piecewise smooth maps, Chaos, Solitons and Fractals, 18 (2003), 953-976. doi: 10.1016/S0960-0779(03)00066-3.
[21]	P. S. Dutta and S. Banerjee, Period increment cascades in a discontinuous map with square-root singularity, Discrete and Continuous Dynamical Systems - Series B, 14 (2010), 961-976. doi: 10.3934/dcdsb.2010.14.961.
[22]	V. Avrutin, M. Schanz and S. Banerjee, Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps, Physical Review E, 75 (2007), 066205. doi: 10.1103/PhysRevE.75.066205.

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