RESEARCH ON BROWNIAN MOVEMENT BASED ON GENERALIZED MANDELBROT–JULIA SETS FROM A CLASS COMPLEX MAPPING SYSTEM
RESEARCH ON BROWNIAN MOVEMENT BASED ON GENERALIZED MANDELBROT–JULIA SETS FROM A CLASS COMPLEX MAPPING SYSTEM
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DOI：
10.1142/s0217984907013560
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发表时间：
2007-08
影响因子：
1.9
通讯作者：
Xing-yuan Wang;W. Liu;Xuejing Yu
中科院分区：
文献类型：
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作者：
Xing-yuan Wang;W. Liu;Xuejing Yu
本文分析了典型的朗之万问题，即带电粒子在双势垒势和含时磁场中运动的几个简单脉冲函数的循环连续影响下的动力学。利用频闪采样，通过选择合适的磁场强度和时间间隔，将朗之万方程简化为一类复映射系统。通过实验数学方法，研究了复映射系统生成的广义M-J(Mandelbrot-Julia)集的结构，阐述了布朗运动理论。作者发现：。(1)通过组合两个简单的复映射，推广了Shirriff构造的M集；(2)广义M-J集的分形结构可以直观地刻画布朗运动的规律，无限重叠的嵌入自相似结构反映了布朗运动的复杂性；(3)所选时间间隔是否显著决定了广义M-J集的分形结构的连续性；(4)质点速度的变化规律取决于相角主值范围的不同选择；(5)如果改变磁场强度和时间间隔的选择，例如选择一个随机波动的磁场，广义J集可能会出现内部填充结构特征，即在速度空间中粒子的不稳定周期轨道的闭合处出现“爆炸”现象。
This article analyzes the typical Langevin problem, i.e. the dynamics of a charged particle under the circularly successive influence of several simple impulse functions moving in a double-well potential and a time-dependent magnetic field. Using the stroboscopic sampling, by selecting an appropriate magnetic intensity and time interval, we reduce the Langevin equation to a class complex mapping system. Through an experimental mathematical method, the authors study the structures of generalized M–J (Mandelbrot–Julia) sets generated by the complex mapping system, and expatiate the theory of Brownian movement. The authors find that:. (1) This paper extends Shirriff constructed M sets by combining two simple complex mappings;. (2) The fractal structure of the generalized M–J sets may visually depict the rule of Brownian movement, and the infinite overlapping embedment self-similar structure reflects the complexity of Brownian movement;. (3) Whether the selected time interval is significant or not determines the continuity of the fractal structure for the generalized M–J sets;. (4) The changing rule of particle velocity depends on the different choices of the principal range of phase angle;. (5) If we change the choices of the magnetic intensity and time interval, for example, we choose a randomly fluctuating magnetic field, the generalized J sets may emerge the interior-filling structure feature, i.e. "explosion" phenomena appear in the closure of the unstable periodic orbits of the particle in the velocity space.