The movement of a particle could be depicted by the Mandelbrot set from the fractal viewpoint. According to the requirement, the movement of the particle needs to show different behaviors. In this paper, the feedback ...The movement of a particle could be depicted by the Mandelbrot set from the fractal viewpoint. According to the requirement, the movement of the particle needs to show different behaviors. In this paper, the feedback control method is taken on the classical Mandelbrot set. By amending the feedback item in the controller, the control method is applied to the generalized Mandelbrot set and by taking the reference item to be the trajectory of another system, the synchronization of Mandelbrot sets is achieved.展开更多
We consider the iterated function system {λz-1, λz + 1} in the complex plane, for A in the open unit disk. Let M be the set of λ such that the attractor of the IFS is connected. We discuss some topological and geom...We consider the iterated function system {λz-1, λz + 1} in the complex plane, for A in the open unit disk. Let M be the set of λ such that the attractor of the IFS is connected. We discuss some topological and geometric properties of the set M and prove a new result about possible corners on its boundary. Some open problems and directions for further research are discussed as well.展开更多
In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analy...In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.展开更多
We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausd...We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.展开更多
In this paper we study the global structure of periodic orbits for a one-dimensionalcomplex map Z_(n+1) =Z_n^m+C by the algebraic analytical method advanced by the authot in1985. We give a general formula for the calc...In this paper we study the global structure of periodic orbits for a one-dimensionalcomplex map Z_(n+1) =Z_n^m+C by the algebraic analytical method advanced by the authot in1985. We give a general formula for the calculation of the orbit number H_N of any period--Norbit. We also verify rigorously that the complex structures of the Mandelbrot set (m =2)and its generalized form (m>2) are composed of infinitely many stable regions of differentperiodic orbits. We find out that the relation between the stable region number I_N of theperiod-N orbit and its orbit number H_N is exactly I_N =N·H_N/m. The algebraic equstionsof the boundary of each element and the locations of its cusp and center can be given pre-cisely. Furthermore, the cause A the infinitely nested structures for these complex figures areexplained.展开更多
Mandelbrot set is self similar, it has a very complex dynamic structure. It is helpful to explore the complex structure of simulating the Mandelbrot set and to calculate its dimension. Problems of precision and veloc...Mandelbrot set is self similar, it has a very complex dynamic structure. It is helpful to explore the complex structure of simulating the Mandelbrot set and to calculate its dimension. Problems of precision and velocity of simulation can be encountered i展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 10971120)the Natural Science Foundation of Shandong Province, China(Grant Nos. ZR2010FM010 and ZR2011FQ035)the Independent Innovation Foundation of Shandong University, China (Grant No. 2011ZRYQ012)
文摘The movement of a particle could be depicted by the Mandelbrot set from the fractal viewpoint. According to the requirement, the movement of the particle needs to show different behaviors. In this paper, the feedback control method is taken on the classical Mandelbrot set. By amending the feedback item in the controller, the control method is applied to the generalized Mandelbrot set and by taking the reference item to be the trajectory of another system, the synchronization of Mandelbrot sets is achieved.
文摘We consider the iterated function system {λz-1, λz + 1} in the complex plane, for A in the open unit disk. Let M be the set of λ such that the attractor of the IFS is connected. We discuss some topological and geometric properties of the set M and prove a new result about possible corners on its boundary. Some open problems and directions for further research are discussed as well.
基金Project supported by the National Basic Research Program (973) of China (Nos. 2004CB719402 and 2002CB312106), the National Natural Science Foundation of China (Nos. 60375020 and 50305033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020335112)
文摘In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.
基金supported by National Natural Science Foundation of China (Grant Nos.10625107, 10801134, 10831004 and 10871047)Natural Science Foundation of Shanghai, China (Grant No.10ZR1403700)
文摘We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.
文摘In this paper we study the global structure of periodic orbits for a one-dimensionalcomplex map Z_(n+1) =Z_n^m+C by the algebraic analytical method advanced by the authot in1985. We give a general formula for the calculation of the orbit number H_N of any period--Norbit. We also verify rigorously that the complex structures of the Mandelbrot set (m =2)and its generalized form (m>2) are composed of infinitely many stable regions of differentperiodic orbits. We find out that the relation between the stable region number I_N of theperiod-N orbit and its orbit number H_N is exactly I_N =N·H_N/m. The algebraic equstionsof the boundary of each element and the locations of its cusp and center can be given pre-cisely. Furthermore, the cause A the infinitely nested structures for these complex figures areexplained.
文摘Mandelbrot set is self similar, it has a very complex dynamic structure. It is helpful to explore the complex structure of simulating the Mandelbrot set and to calculate its dimension. Problems of precision and velocity of simulation can be encountered i
