平流方程拓层为高维动力系统及其多平衡...

A NUMERICAL STUDY ON HIGH-DIMENSIONALITY DYNAMIC SYSTEMS EXTENDED FROM THE ADVECTION EQUATION TOGETHER WITH THE NATURE OF THEIR EQUILIBRIA

本文按一维平流方程,将变量用傅里叶级数展开,取低阶谱导出n=5,6,7等高维动力系统,在动能守恒条件下,研究各动力系统可能存在的平衡态。通过数值试验,讨论了平衡态的性质并判断了稳定性。结果表明:随着维数的增高,各动力系统中可能存在的平衡态并不明显地增多,而稳定的平衡态都仅仅只有2-3个(不包括周期态)。特别有意义的是,有好几个平衡态附近的点会趋向于低维子空间流型上去,且在子流型附近的点也大多不离开这个子流型。子流型的出现是以往所没有注意到的现象。

A study is made of the equilibria possibly available in thedynamic systems under the condition of kinetic-energy conservation interms of one-dimensional advection equation with the variables expandedby Fourier series and then with the low-order spectra to derive high-di-mensionality systems(n=5, 6 and 7). Based on the experimental resultsthe nature of the equilibria is discussed and the stability is decided.Results show that as dimensionality grows higher, the number of equili-bria possibly existing in such systems does not increase remarkably,with only two or three stable equilibria (not including periodic states)available.It is of particular interest that points around a few of theequilibria tend to be in the low-dimensionality flow patterns in subspaceand most of those already around the subpatterns keep staying there. InParticular, the emergence of these subpatterns is a new phenomenon wehave just noticed.