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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 893-897.doi: 10.3969/j.issn.1005-3085.2015.06.010

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Navier-Stokes方程九模类Lorenz方程组的动力学行为及数值仿真

王贺元,   崔  进   

  1. 辽宁工业大学理学院，锦州  121001
  • 收稿日期:2014-07-07 接受日期:2015-05-21 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金 (11572146)；国家重点基础研究发展计划(973计划) (2012CB416605)；辽宁省教育厅科研基金 (L2013248)；锦州市科学技术基金 (13A1D32).

The Dynamical Behavior and Numerical Simulation of a Nine-modes Lorenz Equations of Navier-Stokes Equations

WANG He-yuan,   CUI Jin   

  1. College of Sciences, Liaoning University of Technology, Jinzhou 121001
  • Received:2014-07-07 Accepted:2015-05-21 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (11572146); the National Key Basic Research Project (2012CB416605); the Scientific Research Fund for Education Department of Liaoning Province (L2013248); the Science and Technology Funds of Jinzhou City (13A1D32).

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摘要： 为深入探讨流体流动的稳定性，本文研究了平面不可压缩Navier-Stokes方程九模类Lorenz方程组的动力学行为及数值仿真问题．对平面不可压缩Navier-Stokes方程进行傅里叶展开，采用新的截取模式得到一个九模类Lorenz系统，研究了系统的对称性、耗散性和吸引子的存在性，讨论了该方程组的定常解及其稳定性．基于分岔图与最大Lyapunov指数谱和庞加莱截面以及功率谱，文中阐述并分析了此新型混沌系统的基本动力学行为，仿真分析了系统动力学行为的演化历程，解释了随参数变化系统的不动点、周期态和混沌态等之间转变的物理过程．

关键词: Navier-Stokes方程, 混沌, 分岔, 动力学行为

Abstract:

In order to explore the stability of the flow, we study the nonlinear dynamical behavior and simulation problem of nine-modes Lorenz system for a two-dimensional incompressible Navier-Stokes equations. A new nine-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. The symmetry, dissipation and existence of attractors of the system are studied, and the stationary solution and their stability properties are discussed. Based on numerical simulation results of bifurcation diagram, the Lyapunov exponent spectrum, Poincare section and power spectrum of the system, some basic dynamical behavior of the new system are investigated briefly, the physics process and evolution of the dynamical behavior from fixed point to periodic and chaotic behaviors are presented simultaneously.

Key words: Navier-Stokes equations, chaos, bifurcation, dynamical behavior

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