基于Simulink仿真的标准映射的研究

Research on Standard Map Based on Simulink

人们对于弹跳球模型的研究由来已久,标准映射是弹跳球模型在高弹跳情形下的二维映射,也是研究混沌现象最重要的物理模型之一。以往对于标准映射的研究主要集中在理论分析,而对于标准映射中的混沌产生过程研究较少。首先推导了标准映射的数学模型。之后基于Simulink建立了标准映射的模型,并以此模型为基础,仿真分析了倍周期分叉到混沌的实现途径,给出了周期分叉过程中的不同周期运动的参数区间及混沌出现条件,并计算得到了分叉图,从而计算出Feigen-baum常数,与公认值比较误差很小。最后在混沌带中发现了周期运动,进而揭示了混沌的复杂性。

Bouncing ball problem has been researched for a long time, the standard map is the two dimensional mapping of bouncing ball model when the ball＇s height is greater than amplitude of table, and is the most important model for researching chaos. The previous researches concentrated on analysis of the standard map in theory. The process of chaos in the standard map was researched rarely. The mathematical model of the standard map is derived firstly, and then a model of the standard map based on Simulink is established. The process from periodic bifurcation to chaos, is observed and the bifurcate map is got for calculating the Feigenbaum constant. Lastly the period motion in chaos strips is found which opens out the complexity of the chaos.