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研究平面单位区域内的二次函数的混沌特性, 发现标准二次映射是Li-Yorke混沌的, 也是Devaney混沌的; 在满足一定条件下, 还存在大量的二次函数是混沌的. 一些二次函数可以使用平移与缩放等变换化为标准二次函数,其混沌特性不变; 同时, 对单位区域上的非标准二次函数进行了初步的研究. 通过计算Lyapunov指数以及绘制分岔图等对二次曲线的混沌特性进一步分析, 其中参数变化的分岔图以及混沌曲线控制点的区域分布图等有一定的研究价值. 另外研究表明, 使用多个二次曲线交叉迭代能够产生较好的混沌序列, 该混沌序列可以应用于图像加密等一些实际应用领域.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
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[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 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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