水轮的混沌现象分析及数值仿真

Chaotic phenomena analysis and numerical simulation of water wheel

通过混沌理论分析和数值仿真方法,解释了水轮的混沌旋转现象。研究了旋转水轮对应的类洛伦兹系统的3个平衡点及其局部稳定性,讨论了该混沌系统的全局稳定性及系统耗散性和吸引子的存在性。运用Matlab数学软件对系统的动力学行为进行仿真分析,从分岔图、吸引子、庞加莱截面、返回映射、Lyapnov指数等指标分析该系统混沌的存在性。分析和数值仿真结果表明:当p大于0小于1时,系统存在一个稳定的平衡点;当p大于1小于1.05845时,系统分岔出2个稳定结点;当p大于1.05845小于15时,稳定结点变为稳定焦点;当p大于15时,系统进入混沌状态。从而分析和解释了水轮从静止到均匀旋转、周期型反转及混沌旋转等一系列现象。

By means of chaos theory analysis and numerical simulation,the chaotic rotation of water wheel is explained.Three equilibrium points and their local stability of a lorentz-like system corresponding to a rotating water wheel are studied.The dynamic behavior of the system is simulated and analyzed by using MATLAB mathematical software,and the existence of chaos of the system is analyzed from the indicators such as bipartite graph,attractor,time series,poincare section,return mapping,Lyapnov index,etc.The results of analysis and numerical simulation show that when p is greater than 0 and less than 1,the system has a stable equilibrium point.When p is greater than 1 and less than 1.05845,the system bifurcates two stable nodes.When p is greater than 1.05845 and less than 15,the stable node becomes the stable focus.When p is greater than 15,the system enters a chaotic state.A series of phenomena from stationary to uniform rotation and chaotic rotation are analyzed and explained.