American Institute of Mathematical Sciences

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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

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
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show all references

References:
[1]

1st edition, IEEE, 2001. Google Scholar

[2]

Physica D, 154 (2001), 171-194. doi: 10.1016/S0167-2789(01)00250-0.  Google Scholar

[3]

Physica D, 220 (2006), 127-145. doi: 10.1016/j.physd.2006.07.001.  Google Scholar

[4]

Journal of Sound and Vibration, 145 (1991), 279-297. doi: 10.1016/0022-460X(91)90592-8.  Google Scholar

[5]

Physical Review E, 55 (1997), 266-270. doi: 10.1103/PhysRevE.55.266.  Google Scholar

[6]

Physica D, 136 (2000), 280-302. doi: 10.1016/S0167-2789(99)00161-X.  Google Scholar

[7]

Physica D, 170 (2002), 170-175.  Google Scholar

[8]

Nonlinearity, 11 (1998), 858-890.  Google Scholar

[9]

IEEE Transactions on Circuits and Systems-I, 52 (2005), 366-378. doi: 10.1109/TCSI.2004.841595.  Google Scholar

[10]

IEEE Transactions on Circuits and Systems-I, 47 (2000), 1160-1177. doi: 10.1109/81.873871.  Google Scholar

[11]

International Journal of Bifurcation and Chaos, 13 (2003), 3341-3352. doi: 10.1142/S0218127403008533.  Google Scholar

[12]

Physics Letters A, 337 (2005), 87-92. doi: 10.1016/j.physleta.2005.01.046.  Google Scholar

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Discrete and Continuous Dynamical Systems - Series B, 13 (2010), 249-267. doi: 10.3934/dcdsb.2010.13.249.  Google Scholar

[14]

Physica D, 57 (1992), 39-57. doi: 10.1016/0167-2789(92)90087-4.  Google Scholar

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International Journal of Bifurcation and Chaos, 5 (1995), 189-207. doi: 10.1142/S0218127495000156.  Google Scholar

[16]

Chaos, Solitons and Fractals, 10 (1999), 1881-1908.  Google Scholar

[17]

IEEE Transactions on Circuits and Systems-I, 47 (2000), 389-394. doi: 10.1109/81.841921.  Google Scholar

[18]

Physical Review E, 59 (1999), 4052-4061. doi: 10.1103/PhysRevE.59.4052.  Google Scholar

[19]

Physical Review E, 49 (1994), 1073-1076. doi: 10.1103/PhysRevE.49.1073.  Google Scholar

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Chaos, Solitons and Fractals, 18 (2003), 953-976. doi: 10.1016/S0960-0779(03)00066-3.  Google Scholar

[21]

Discrete and Continuous Dynamical Systems - Series B, 14 (2010), 961-976. doi: 10.3934/dcdsb.2010.14.961.  Google Scholar

[22]

Physical Review E, 75 (2007), 066205. doi: 10.1103/PhysRevE.75.066205.  Google Scholar

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