Homogenization Method for the Quadratic Polynomial Chaotic System
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摘要: 该文给出了一般的2次多项式混沌系统与Tent映射拓扑共轭的充分条件，并依据该条件，给出了一类2次多项式混沌系统及其概率密度函数；进一步得到了能够将这类系统均匀化的变换函数；给出了一个新的2次多项式混沌系统并进行均匀化处理，对其产生的序列进行了信息熵、Kolmogorov熵和离散熵分析，结果显示该均匀化方法的均匀化效果显著且不改变序列混沌程度。Abstract: A sufficient condition for general quadratic polynomial systems to be topologically conjugate with Tent map is proposed. Base on this condition, the probability density function of a class of quadratic polynomial systems is provided and transformations function which can homogenize this class of chaotic systems is further obtained. The performances of both the original system and the homogenized system are evaluated. Numerical simulations show that the information entropy of the uniformly distributed sequences is closer to the theoretical limit and its discrete entropy remains unchanged. In conclusion, with such homogenization method all the chaotic characteristics of the original system is inherited and better uniformity is performed.
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Key words:
- Chaotic system /
- Homogenization /
- Topologically conjugate /
- Entropy
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表 1 几个2次多项式混沌系统
混沌系统$f(x)$ 概率密度 均匀化系统$z(x)$ $f(x)$信息熵 $z(x)$信息熵 $f(x) = \frac{7}{2}{x^2} + \frac{{33}}{{10}}x - \frac{{53}}{{200}}$ $\frac{7}{{{\text{π}} \sqrt { - 49{x^2} + 46.2x + 5.11} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin \left( { - \frac{7}{4}x - \frac{{33}}{{40}}} \right)$ 8.6470 8.9651 $f(x) = \frac{5}{4}{x^2} - \frac{1}{2}x - \frac{{27}}{{20}}$ $\frac{5}{{{\text{π}} \sqrt { - 25{x^2} + 10x + 63} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin\left( { - \frac{5}{8}x + \frac{1}{8}} \right)$ 8.6380 8.9649 $f(x) = - \frac{5}{2}{x^2} + 3x + \frac{1}{2}$ $\frac{5}{{{\text{π}} \sqrt { - 25{x^2} + 30x + 7} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin\left( {\frac{5}{4}x - \frac{3}{4}} \right)$ 8.6412 8.9650 
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表 2 系统均匀化前后的信息熵与最大熵比较
$\left( {N,M} \right)$ 均匀化前
信息熵均匀化后
信息熵最大熵 (500000, 100) 6.3530 6.6437 6.6439 (500000, 300) 7.9137 8.2284 8.2288 (500000, 500) 8.6431 8.9651 8.9658
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