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In the paper, the chaotic characteristic of the quadratic function in plane unit area is studied, and it is found that the standard quadratic mapping is Li-Yorke chaos, and also Devaney chaos, and that under certain conditions, there are a large number of quadratic functions that are chaotic. Some quadratic functions can transform into the standard quadratic functions by moving and zooming, without changing their chaotic characteristics. In addition, non-standard quadratic function is preliminary studied. The chaotic characteristic of the quadratic curve is analyzed by calculating Lyapunov exponents and drawing the bifurcation diagram of conic. The bifurcation diagram of the parameter variation and the area distributing diagram of parameter control points have certain research value. The study also shows that more conic curve cross iteration can generate a better chaotic sequence, and the chaotic sequence can be used to image encryption and other practical purposes.
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Keywords:
- chaos /
- conic /
- plane of unit area
[1] Chen Y C 2008 Int. J. Bifurc. Chaos 18 1825
[2] Góra P, Boyarsky A 2003 Int. J. Bifurc. Chaos 13 1299
[3] Lai D, Chen G 2000 Int. J. Bifurc. Chaos 10 1509
[4] 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 物理学报 60 010504]
[5] Li C, Chen G 2003 Chaos, Solitons and Fractal 18 807
[6] Chen X F, Chen G, Yu X 2000 Chaos, Solitons and Fractal 10 771
[7] Chen G, Huang T, Huang Y 2004 Int. J. Bifurc. Chaos 14 2161
[8] Zhang X, Shi Y, Chen G 2009 Int. J. Bifurc. Chaos 19 531
[9] Li T Y, York J A 1975 American Mathematical Monthly 82 985
[10] Banks J, Brooks J C 1992 American Mathematical Monthly 99 332
[11] Wang L, Huang G, Huan S 2007 Nonlinear Anal. 67 2131
[12] Liu H, Wang L D, Chu Z Y 2009 Nonlinear Anal. 71 6144
[13] Akhmet M U 2009 Math. Anal. Appl. 351 804
[14] Pisarchik A N, Zanin M 2008 Physica D: Monlinear Phenomena 237 2638
[15] Gao T, Chen Z 2008 Phys. Lett. A 372 394
[16] Huang C K, Nien H H 2009 Opti. Comm. 282 2123
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[1] Chen Y C 2008 Int. J. Bifurc. Chaos 18 1825
[2] Góra P, Boyarsky A 2003 Int. J. Bifurc. Chaos 13 1299
[3] Lai D, Chen G 2000 Int. J. Bifurc. Chaos 10 1509
[4] 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 物理学报 60 010504]
[5] Li C, Chen G 2003 Chaos, Solitons and Fractal 18 807
[6] Chen X F, Chen G, Yu X 2000 Chaos, Solitons and Fractal 10 771
[7] Chen G, Huang T, Huang Y 2004 Int. J. Bifurc. Chaos 14 2161
[8] Zhang X, Shi Y, Chen G 2009 Int. J. Bifurc. Chaos 19 531
[9] Li T Y, York J A 1975 American Mathematical Monthly 82 985
[10] Banks J, Brooks J C 1992 American Mathematical Monthly 99 332
[11] Wang L, Huang G, Huan S 2007 Nonlinear Anal. 67 2131
[12] Liu H, Wang L D, Chu Z Y 2009 Nonlinear Anal. 71 6144
[13] Akhmet M U 2009 Math. Anal. Appl. 351 804
[14] Pisarchik A N, Zanin M 2008 Physica D: Monlinear Phenomena 237 2638
[15] Gao T, Chen Z 2008 Phys. Lett. A 372 394
[16] Huang C K, Nien H H 2009 Opti. Comm. 282 2123
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