Acta mathematica scientia,Series A ›› 2017, Vol. 37 ›› Issue (1): 199-216.

Previous Articles    

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

Wang Heyuan   

  1. College of Sciences, Liaoning University of Technology, Liaoning Jinzhou 121001
  • Received:2016-07-10 Revised:2016-12-02 Online:2017-02-26 Published:2017-02-26
  • Supported by:

    Supported by the NSFC (11572146, 11526105), the Funds for Education Department of Liaoning Province (L2013248) and the Science and Technology Funds of Jinzhou City (13A1D32)

Abstract:

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, Poincare 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).

Key words: Hopf bifurcation, Lyapunov function, Strange Attractors, Chaos

CLC Number: 

  • O175.14
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