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Border-collision bifurcations in a generalized piecewise linear-power map
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 |
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. |
show all references
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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