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A new two-dimensional sinusoidal discrete map is achieved by nonlinearly coupling a sinusoidal map and with a cubic map. The fixed points and the corresponding eigenvalues are obtained based on this two-dimensional sinusoidal discrete map, and the stability of the system is analyzed to study the complex nonlinear dynamic behavior of the system and the evolutions of their attractors. The research results indicate that there are complex nonlinear physical phenomena in this two-dimensional sinusoidal discrete map, such as symmetry breaking bifurcation, Hopf bifurcation, period doubling bifurcation, periodic oscillation fast-slow effect, etc. Furthermore, bifurcation mode coexisting, fast-slow periodic oscillations and the evolutions of the attractors of the system are analyzed by using the bifurcation diagram, the Lyapunov exponent diagram and the phase portraits when the control parameters of the system are varied, and the correctness of the theoretical analysis is verified based on numerical simulations.
[1] Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502
[2] Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[3] Parui S, Banerjee S 2003 IEEE Trans. Circuits Syst. I 50 1464
[4] Yan H, Wei P, Xiao X C 2010 Chin. Phys. B 19 090501
[5] Yan H, Wei P, Xiao X C 2009 Chin. Phys. B 18 3287
[6] Van Ha N, Han J S 2013 Chin. Phys. Lett. 30 060501
[7] Wu J, Zhan X S, Zhang X H, Gao H L 2012 Chin. Phys. Lett. 29 050203
[8] Namajunas A, Pyragas K, Tamasevicius A 1997 Int. J. Bifurc. Chaos 7 957
[9] Wu X, Tse C K, Dranga O, Lu J 2006 IEEE Trans. Circuits Syst. I 53 204
[10] Yang R, Zhang B 2007 Chin. Soc. Elec. Eng. 27 114 (in Chinese) [杨汝, 张波 2007 中国电机工程学报 27 114]
[11] Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 020504
[12] Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 080506
[13] Boyland P L 1986 Comm. Math. Phys. 106 353
[14] Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[15] Hu W, Zhao G H, Zhang G, Zhang J Q, Liu X L 2012 Acta Phys. Sin. 61 170505 (in Chinese) [胡文, 赵广浩, 张弓, 张景乔, 刘贤龙 2012 61 170505]
[16] Zhang H, Chu Y D, Ding W C, Li X F 2013 Acta Phys. Sin. 62 040202 (in Chinese) [张惠, 褚衍东, 丁旺才, 李险峰 2013 62 040202]
[17] Bao B C, Kang Z S, Xu J P, Hu W 2009 Acta Phys. Sin. 58 1420 (in Chinese) [包伯成, 康祝圣, 许建平, 胡文 2009 58 1420]
[18] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 60 010504]
[19] Edward O 2002 Chaos in Dynamical System (Cambridge: Cambridge University Press) p50
[20] Zhou G H, Xu J P, Bao B C, Zhang F, Liu X S 2010 Chin. Phys. Lett. 27 090504
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[1] Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502
[2] Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[3] Parui S, Banerjee S 2003 IEEE Trans. Circuits Syst. I 50 1464
[4] Yan H, Wei P, Xiao X C 2010 Chin. Phys. B 19 090501
[5] Yan H, Wei P, Xiao X C 2009 Chin. Phys. B 18 3287
[6] Van Ha N, Han J S 2013 Chin. Phys. Lett. 30 060501
[7] Wu J, Zhan X S, Zhang X H, Gao H L 2012 Chin. Phys. Lett. 29 050203
[8] Namajunas A, Pyragas K, Tamasevicius A 1997 Int. J. Bifurc. Chaos 7 957
[9] Wu X, Tse C K, Dranga O, Lu J 2006 IEEE Trans. Circuits Syst. I 53 204
[10] Yang R, Zhang B 2007 Chin. Soc. Elec. Eng. 27 114 (in Chinese) [杨汝, 张波 2007 中国电机工程学报 27 114]
[11] Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 020504
[12] Xu J, Long K P, Fournier-Prunaret D, Taha A K, Charge P 2010 Chin. Phys. Lett. 27 080506
[13] Boyland P L 1986 Comm. Math. Phys. 106 353
[14] Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[15] Hu W, Zhao G H, Zhang G, Zhang J Q, Liu X L 2012 Acta Phys. Sin. 61 170505 (in Chinese) [胡文, 赵广浩, 张弓, 张景乔, 刘贤龙 2012 61 170505]
[16] Zhang H, Chu Y D, Ding W C, Li X F 2013 Acta Phys. Sin. 62 040202 (in Chinese) [张惠, 褚衍东, 丁旺才, 李险峰 2013 62 040202]
[17] Bao B C, Kang Z S, Xu J P, Hu W 2009 Acta Phys. Sin. 58 1420 (in Chinese) [包伯成, 康祝圣, 许建平, 胡文 2009 58 1420]
[18] Meng J D, Bao B C, Xu Q 2011 Acta Phys. Sin. 60 010504 (in Chinese) [孟继德, 包伯成, 徐强 2011 60 010504]
[19] Edward O 2002 Chaos in Dynamical System (Cambridge: Cambridge University Press) p50
[20] Zhou G H, Xu J P, Bao B C, Zhang F, Liu X S 2010 Chin. Phys. Lett. 27 090504
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