American Institute of Mathematical Sciences

Advanced
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

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 & 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, (2001).   Google Scholar

[2]

M. di Bernardo, C. J. Budd and A. R Champneys, Corner collision implies border-collision bifurcation,, Physica D, 154 (2001), 171.  doi: 10.1016/S0167-2789(01)00250-0.  Google Scholar

[3]

P. T. Piiroinen and C. J. Budd, Corner bifurcations in non-smoothly forced impact oscillators,, Physica D, 220 (2006), 127.  doi: 10.1016/j.physd.2006.07.001.  Google Scholar

[4]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator,, Journal of Sound and Vibration, 145 (1991), 279.  doi: 10.1016/0022-460X(91)90592-8.  Google Scholar

[5]

A. B. Nordmark, Universal limit mapping in grazing bifurcations,, Physical Review E, 55 (1997), 266.  doi: 10.1103/PhysRevE.55.266.  Google Scholar

[6]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Physica D, 136 (2000), 280.  doi: 10.1016/S0167-2789(99)00161-X.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TCSI.2004.841595.  Google Scholar

[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.  doi: 10.1109/81.873871.  Google Scholar

[11]

P. Jain and S. Banerjee, Border collision bifurcations in one-dimensional discontinuous maps,, International Journal of Bifurcation and Chaos, 13 (2003), 3341.  doi: 10.1142/S0218127403008533.  Google Scholar

[12]

A. Kumar, S. Banerjee and D. P. Lathrop, Dynamics of a piecewise smooth map with sigularity,, Physics Letters A, 337 (2005), 87.  doi: 10.1016/j.physleta.2005.01.046.  Google Scholar

[13]

G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps,, Discrete and Continuous Dynamical Systems - Series B, 13 (2010), 249.  doi: 10.3934/dcdsb.2010.13.249.  Google Scholar

[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.  doi: 10.1016/0167-2789(92)90087-4.  Google Scholar

[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.  doi: 10.1142/S0218127495000156.  Google Scholar

[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, 10 (1999), 1881.   Google Scholar

[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.  doi: 10.1109/81.841921.  Google Scholar

[18]

S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps,, Physical Review E, 59 (1999), 4052.  doi: 10.1103/PhysRevE.59.4052.  Google Scholar

[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.  doi: 10.1103/PhysRevE.49.1073.  Google Scholar

[20]

C. Halse, M. Homer and M. di Bernardo, C-bifurcation and period-adding in one-dimensional piecewise smooth maps,, Chaos, 18 (2003), 953.  doi: 10.1016/S0960-0779(03)00066-3.  Google Scholar

[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.  doi: 10.3934/dcdsb.2010.14.961.  Google Scholar

[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).  doi: 10.1103/PhysRevE.75.066205.  Google Scholar

show all references

References:
[1]

S. Banerjee and G. C. Verghese, "Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations,", 1st edition, (2001).   Google Scholar

[2]

M. di Bernardo, C. J. Budd and A. R Champneys, Corner collision implies border-collision bifurcation,, Physica D, 154 (2001), 171.  doi: 10.1016/S0167-2789(01)00250-0.  Google Scholar

[3]

P. T. Piiroinen and C. J. Budd, Corner bifurcations in non-smoothly forced impact oscillators,, Physica D, 220 (2006), 127.  doi: 10.1016/j.physd.2006.07.001.  Google Scholar

[4]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator,, Journal of Sound and Vibration, 145 (1991), 279.  doi: 10.1016/0022-460X(91)90592-8.  Google Scholar

[5]

A. B. Nordmark, Universal limit mapping in grazing bifurcations,, Physical Review E, 55 (1997), 266.  doi: 10.1103/PhysRevE.55.266.  Google Scholar

[6]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Physica D, 136 (2000), 280.  doi: 10.1016/S0167-2789(99)00161-X.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TCSI.2004.841595.  Google Scholar

[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.  doi: 10.1109/81.873871.  Google Scholar

[11]

P. Jain and S. Banerjee, Border collision bifurcations in one-dimensional discontinuous maps,, International Journal of Bifurcation and Chaos, 13 (2003), 3341.  doi: 10.1142/S0218127403008533.  Google Scholar

[12]

A. Kumar, S. Banerjee and D. P. Lathrop, Dynamics of a piecewise smooth map with sigularity,, Physics Letters A, 337 (2005), 87.  doi: 10.1016/j.physleta.2005.01.046.  Google Scholar

[13]

G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps,, Discrete and Continuous Dynamical Systems - Series B, 13 (2010), 249.  doi: 10.3934/dcdsb.2010.13.249.  Google Scholar

[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.  doi: 10.1016/0167-2789(92)90087-4.  Google Scholar

[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.  doi: 10.1142/S0218127495000156.  Google Scholar

[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, 10 (1999), 1881.   Google Scholar

[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.  doi: 10.1109/81.841921.  Google Scholar

[18]

S. Banerjee and C. Grebogi, Border collision bifurcations in two-dimensional piecewise smooth maps,, Physical Review E, 59 (1999), 4052.  doi: 10.1103/PhysRevE.59.4052.  Google Scholar

[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.  doi: 10.1103/PhysRevE.49.1073.  Google Scholar

[20]

C. Halse, M. Homer and M. di Bernardo, C-bifurcation and period-adding in one-dimensional piecewise smooth maps,, Chaos, 18 (2003), 953.  doi: 10.1016/S0960-0779(03)00066-3.  Google Scholar

[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.  doi: 10.3934/dcdsb.2010.14.961.  Google Scholar

[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).  doi: 10.1103/PhysRevE.75.066205.  Google Scholar

[1]

Iryna Sushko, Anna Agliari, Laura Gardini. Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 881-897. doi: 10.3934/dcdsb.2005.5.881
[2]

Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249
[3]

Laura Gardini, Roya Makrooni, Iryna Sushko. Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 701-729. doi: 10.3934/dcdsb.2018039
[4]

Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014
[5]

Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417
[6]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[7]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[8]

Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493
[9]

Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289
[10]

Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 99-130. doi: 10.3934/dcds.1998.4.99
[11]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[12]

Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529
[13]

Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287
[14]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[15]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
[16]

Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127
[17]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103
[18]

Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779
[19]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[20]

Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences