厄农映射

厄农映射（英语：Hénon map）是一种可以产生混沌现象的离散时间动态系统，其迭代表达式为：

a = 1.4、b = 0.3时的厄农吸引子

{\begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\y_{n+1}=bx_{n}.\end{cases}}

在经典厄农映射中，参数值分别取为a = 1.4及b = 0.3。此时，系统表现出混沌现象。而当a与b取其他不同值时，系统可表现为混沌现象、阵发性现象，或收敛至周期点。通过轨道图可以看出不同参数下系统的行为特征。

厄农映射是由法国数学家米歇尔·厄农（英语：Michel Hénon）提出的，以此作为洛伦茨模型的庞加莱截面的简化模型。对经典厄农映射而言，任意初始点或趋向厄农奇异吸引子，或发散至无穷大。厄农吸引子具有分形结构，其在一个方向上连续，另一个方向上则为一个康托尔集。数值计算表明经典厄农吸引子的关联维数为1.25 ± 0.02^[1]，豪斯多夫维数为1.261 ± 0.003。^[2]

参考文献

^ P. Grassberger; I. Procaccia. Measuring the strangeness of strange attractors. Physica. 1983, 9D (1-2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.
^ D.A. Russell; J.D. Hanson; E. Ott. Dimension of strange attractors. Physical Review Letters. 1980, 45 (14): 1175. Bibcode:1980PhRvL..45.1175R. doi:10.1103/PhysRevLett.45.1175.

M. Hénon. A two-dimensional mapping with a strange attractor. Communications in Mathematical Physics. 1976, 50 (1): 69–77. Bibcode:1976CMaPh..50...69H. doi:10.1007/BF01608556.
Predrag Cvitanović; Gemunu Gunaratne; Itamar Procaccia. Topological and metric properties of Hénon-type strange attractors. Physical Review A. 1988, 38 (3): 1503–1520. Bibcode:1988PhRvA..38.1503C. PMID 9900529. doi:10.1103/PhysRevA.38.1503.
M. Michelitsch; O. E. Rössler. A New Feature in Hénon's Map. Computers & Graphics. 1989, 13 (2): 263–265 [2016-12-03]. doi:10.1016/0097-8493(89)90070-8. （原始内容存档于2021-01-25）. . Reprinted in: Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 69–71, 1998

[Grassberger_1983-1] P. Grassberger; I. Procaccia. Measuring the strangeness of strange attractors. Physica. 1983, 9D (1-2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.

[Russell_1980-2] D.A. Russell; J.D. Hanson; E. Ott. Dimension of strange attractors. Physical Review Letters. 1980, 45 (14): 1175. Bibcode:1980PhRvL..45.1175R. doi:10.1103/PhysRevLett.45.1175.

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