构造新分形生成算法对复杂指数广义M集的研究

Mandelbrot set is called M set for short. A k-M set is constructed when the exponent in iteration function is defined as 'k'. A k-M set is called generalized M set. Generalized M set and fractal generation algorithm are widely applied into many research domains, such as chaotic system and complex system..Today, there are two problems in this area, which are a) there is no quantitative study of k-M sets when k is a real or complex number; b) the correctness and effectiveness of exist universal algorithms are shown bad performance. So in this project, we construct a novel fractal generation algorithm and apply this algorithm into the study of generalized M set with complex exponent by combined computer graphics, fractal theory, function of complex variable, and complex analytic dynamics..First, we have the asymptote, convergent and divergent area of fractal set. Then, a novel fractal generation algorithm is created by combined both asymptote family, convergence and divergence. We also extend the algorithm to distributed environment. Moreover, we study the characteristic of k-M set when k is a real number by applied both this novel algorithm and rational approximation. Finally, we study k-M set when k is a complex number by extended the research result of k- M set when k is a rational number. We separately study the effect induced by real and imaginary part of exponent, and the effect induced by real part in coordination with imaginary part of exponent..In conclusion, this project presents a novel fractal generation algorithm and provides a novel way to solve problems in k-M set by applied this novel algorithm. The study of this project can improve fundamental research of both generalized M set and fractal generation algorithm.

Mandelbrot集，简称M集，将其迭代函数指数扩展后构成广义M集。广义M集和分形生成算法目前已广泛应用在混沌系统、复杂系统等相关领域中。虽然对广义M集的研究很多，但目前的研究存在下面几点问题：（1）尚未有对实数阶和复数阶广义M集的定量研究；（2）现有的普适分形生成算法的正确性和效率难于保证。.因此本项目结合图形学、分形论、复变理论与复解析动力学的知识构造新的分形生成算法，并使用其对复杂指数广义M集进行研究。首先通过区域中点迭的代特征得到分形集的敛散区域和渐近线；然后结合这两者构造新的分形生成算法，并向分布式系统进行推广；再后使用新的分形生成算法和有理数的逼近对实数阶广义M集的分形性质研究；最后从指数的实部、虚部分别研究其对广义M集的影响，并研究其协同影响。.本项目提出一种新的分形生成算法，并使用这种算法对广义M集的研究提出新的解决方法，可以进一步推动广义M集和分形的基础研究。

Mandelbrot集，简称M集，将其迭代函数指数扩展后构成广义M集，目前的研究存在下面两点问题：（1）尚未有对实数和复数阶广义M集的定量研究；（2）现有的普适分形生成算法的正确性和效率难于保证。.本项目结合了图形学、分形论、复变函数理论与复解析动力学的知识构造了新的分形生成算法，并使用其对复杂指数广义M集进行了研究。本项目（1）得到了广义M集的敛散区域和渐近线；（2）结合敛散性和渐近线，结合并行理论构造新的分形生成算法；（3）使用新的分形生成算法对实数阶广义M集的分形性质进行了研究；（4）对复数阶广义M集进行了研究。.随后，本项目利用提出的新理论和方法，对复杂信息中的分形机理和特征进行了简单研究，并发现复杂信息中的分形特征具有很强的平滑性和鲁棒性，并与熵有着某种程度上的联系。可以预见的是，对这些工作的进一步研究可以进一步推动分形和信息科学的基础和应用研究。