Research On Fractal Characters Of Generalized 3x+1 Functions And Generalized Mandelbrot Set

For generalized 3x+1 function T(x), the feature of fixed points and their existence-domain analysis is an important problem in fractal. T(x) is a complex transcendental function and its fixed point in C-plane is hardly to solve. Meanwhile, the feature of fixed point is difficult to analyze; all these become an obstacle for the further study of T(x) dynamic system.In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). We create a generalized approximate 3x+1 function B(x). We analyze B(x) in real axis and C-plane and find the fixed point of B(x) at real axis and C-plane. Then we prove zero is the only attract fixed point. Moreover, we show the symmetry of B(x) and use escape time algorithm to draw the fractal figures. We show B(x)'s iteration have endless points make (?) B~i(x)=∞in [n,n+1] when n∈and |n|>3. At last, we put forward a conjecture that all the divergence areas of B(x) is similar to each other. The research conclusion of B(x) is play a direct hole in promoting research of T(x).In fact, by research conclusion of B(x), we study more of the characters of T(x)'s fixed point. Firstly, Because of the topological invariance of T(x), the constructive proof of its fixed point in C-plane is presented, and then the analysis for the existence domain of fixed points is given as well as their feature. Based on the existence domain of T(x) fixed points, their distributions in C-plane are estimated. So a numerical algorithm for solving the fixed points of T(x) is put forward on the basis of distributions analyses. Furthermore, some convergence domains of T(x) in C-plane are obtained, as well as fractal image of these domains. The result of numerical experiment shows that the algorithm in this paper is correct and easy to implement.Moreover, In order to study more about the fractal character of T(x), we research the characters of T(x)'s periodic points with the research of T(x)'s fixed point. In this paper, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point's number and proved it. Moreover, we found the number of T(x)'s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.At the same time, we also study fractal construction of another generalized 3x+1 function C(x). At first, we find the character of fixed points of C(x) at real axis by complex analytical analysis. Then we improve the solving algorithm of its fixed points. We proved that the integer fixed points of C(x) are 0 and -1 and the attract fixed points of C(x) are 0 and -1.2777. Popularized the result to complex plane, we prove that there is no fixed point of C(z) except real axis. We draw the fractal figures of C(x) by escape time algorithm to prove the result and give a conjecture from the fractal figures. The conjecture guess that the attract domain of C(x) is a connected domain which is connected with single point. Then we study the fractal structure of generalized M set in this paper. It is well known that Mandelbrot set is an important part of fractal and chaos for its simple expression and complex structure, In this essay, we construct a class of polynomial curves and gain boundary points and periodic points of M-set. At first, we have proved that the threshold is number 2. In other words, for point z in complex plane, it is not in M-set when |fn(z)|>2 (fn(z)=f○fn-1(z)). Secondly, with this threshold condition, we construct a class of polynomial approximation curves for M-set. We have proved that the curves are contained level by level, and the limit curve is M-Set when the iteration tends to infinity. Then, we construct the equations to gain periodic and boundary points by using the polynomial approximation curves. Finally, we extend the conclusion to the generalized M-Set with positive integer order, and prove that there exists similar conclusion for the generalized M-Set with positive integer order.Otherwise, we research generalized M set with negative integer index. We define a series {yi} as the ith iterations of generalized M set. By to study characters of each item of series {yi-y1}, we solve all escape points of its fractal image and point out that all these points are denumerable infinite and discrete. Then we put forward a better algorithm to create figures more accurate. To compare with the different points in fractal figures drawn by classic and our algorithm, we find the new algorithm draws better figures to fit the definition of M set.Further more, in order to create the fractal image faster, we improve the fractal image creating algorithm. It is known that the escape time algorithm is the most universal algorithm when to create fractal image. With the study of a class of algorithms based on escape time algorithm, we find there is wasting-calculation in these algorithms. In this paper, when combined with the feature of eventually periodic point of functions, we define a kind of points as no-escape point. To analyze the shortcomings of the classic algorithm, we improve the escape time algorithm base on the no-escape points. We analyze the algorithm and put forward the best application scope for it. By create lots of fractal figures, we find the figures created by the two algorithms are consistent with each other except a few escape points. We compare the complexity between the two algorithms and find the iteration times by the improved algorithm are less than escape time algorithm when creating the fractal figures. We do several experiments and find the improved algorithm is universal and it reduces the time-consuming.