The second largest Lyapunov exponent and transition to chaos of natural convection in a rectangular cavity
The second largest Lyapunov exponent and transition to chaos of natural convection in a rectangular cavity
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DOI：
10.1016/j.ijheatmasstransfer.2005.06.046
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发表时间：
2006-12
影响因子：
5.2
通讯作者：
H. Ishida;S. Kawase;H. Kimoto
中科院分区：
文献类型：
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作者：
H. Ishida;S. Kawase;H. Kimoto
在这项研究中，我们提出了一种准确而简单的方法来评估李雅普诺夫谱。该方法适用于任何最终将控制方程组表示为常微分方程组的离散化方法。将该方法应用于具有加热和冷却侧壁的矩形空腔内自然对流的李雅普诺夫指数的计算。主要结果如下：(1)最大李雅普诺夫指数和第二大李雅普诺夫指数可以在没有任何影响指数的参数的情况下计算。(2)利用第二大Lyapunov指数，可以根据瑞利数将热对流场定量地划分为五个区域，并通过识别第一和第二个Hopf分叉来阐明从稳态到混沌的转变路径。(3)热对流场在发生Hopf分叉的临界Rayleigh数以上的起伏可以用归一化的Lyapunov向量来定量解释，而归一化的Lyapunov向量结合Lyapunov指数的计算，恰好在临界点以下。
In this study we have proposed an accurate and simple method to evaluate the Lyapunov spectrum. The method is suitable for any discretization method that finally expresses a governing equation system in the form of an ordinary differential equation system. The method was applied to evaluate up to the second largest Lyapunov exponents for natural convection in a rectangular cavity with heated and cooled side walls. The main results are as follows: (1) the largest and second largest Lyapunov exponents can be evaluated without any parameters that affects the exponents. (2) The second largest Lyapunov exponent makes it possible to classify quantitatively thermal convection fields into five regimes against the Rayleigh number and to clarify the transition route from steady state to chaos by identifying the first and second Hopf bifurcations. (3) The fluctuation in thermal convection fields just over the critical Rayleigh number at which Hopf bifurcation occurs can be quantitatively explained by using normalized Lyapunov vectors, associated with the computation of the Lyapunov exponents, just under the critical point.