平面不可压缩磁流体动力学五模类Lorenz方程组的动力学行为及其数值仿真  

The Dynamical Behaviors and the Numerical Simulation of a Five-Mode Lorenz-Like System of the MHD Equations for a Two-Dimensional Incompressible Fluid on a Torus

作　　者：王贺元[1] 

机构地区：[1]辽宁工业大学理学院,辽宁锦州121001

出　　处：《数学物理学报（A辑）》2017年第1期199-216,共18页Acta Mathematica Scientia

基　　金：国家自然科学基金(11572146;11526105);辽宁省教育厅科研基金(L2013248);锦州市科技专项基金(13A1D32)~~

摘　　要：该文研究了平面正方形区域上不可压缩的磁流体动力学方程组五模截断所得到的十维模型的动力学行为问题.首先,利用模式截断方法推导了十模系统,讨论了该方程组定常解及其稳定性,其次,发现了Hopf分叉和混沌,证明了该方程组吸引子的存在性和全局稳定性,最后,给出了系统从分叉到混沌整个过程所呈现的动力学行为演变的详细数值模拟结果,分析了磁性对系统动力学行为的影响.基于分岔图、Lyapunov指数谱和庞加莱截面图,返回映射和功率谱等数值模拟结果揭示了这个低维系统的动力学行为特征.这个新混沌系统通过周期倍分岔过渡到混沌(费根鲍姆途径).In this paper, we investigate a ten-dimensional model of plane flow, obtained by a five-mode truncation of the magnetic hydrodynamic （MHD） equations for a two-dimensional incompressible fluid on a torus. First, we derive a model system by taking five modes in the Fouriers expansion, and then discuss the stability of stationary solutions of the five-mode system. Second, we find phenomena of Hopf bifurcation and chaos, and prove the existence of its attractor and global stability of the five-mode system. Finally, we present a detailed numerical result of the whole process from bifurcation to chaos, and analyze the influence of magnetism on the dynamical behavior of the system. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poilmare section, power spectrum and return map of the system, some basic dynamical behavior of this low-dimensional model are revealed, the new chaos system exhibits a route to chaos via a period doubling cascade （Feigenbaum＇s Scenario）.

关 键 词：HOPF分歧 李雅普诺夫函数 奇怪吸引子 混沌 

分 类 号：O175.14[理学—数学] O241.82[理学—基础数学]