非稳态非Darcy渗流系统的混沌行为研究
首发时间:2010-02-04
摘要:将水在岩层中一维非稳态、非Darcy渗流系统简化为含5个控制参量和6个状态变量的非线性常微分方程组描述的系统,通过响应计算和时间序列分析,讨论了渗流系统的混沌行为:首先计算了降阶系统在某组控制参量下响应,利用快速Fourier变换计算了吸引子的功率谱;其次通过相空间重构,根据关联积分-覆盖半径对数曲线计算了吸引子的关联维数;最后运用Gram-Schmit正交化方法计算了吸引子的Lyapunov指数。结果表明,吸引子的功率谱是连续的,关联维数等于2.36,Lyapunov指数 LE1、LE2、LE3为正值,LE5、LE6为负值,而LE4在零附近波动。根据吸引子功率谱、关联维数和Lyapunov指数三方面的特征,可以判定非稳态、非Darcy渗流系统出现混沌。
关键词: 非Darcy流 混沌 功率谱 关联维数 Lyapunov指数
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Study on Chaotic Behaviour of Non-steady and Non-Darcy Flow System
Abstract:One dimensinal non-steady, non-Darcy flow of water in a rock stratum was reduced into a system described by 6 ordinary differential equations involving 5 controlling parameters. Through response computations and time series analysis, chaotic behavior in the reduced system was discussed in details. Firstly, the dynamical response of the reduced system under a set of parameters was calculatted, and the power spectrum of the attractor was obtained through fast Lagrangian transformation; then through phase space reconstruction, the correlation dimension of the attractor was calculated according to the logarithmic curves of correlation integral vs. covering radius; and lastly, the Lyapunov indices of the attractor were calculatted by using Gram-Schmit’s orthogonalization method. The results show that the power spectrum of the attractor is continuous; the correlation dimension of the attractor is equal to 2.36; among the Lyapunov indices, LE1, LE2, LE3 are positive, LE5, LE6 are negative, and LE4 fluctuates near zero. All the analysis indicates that there may exist chaos in the system of non-steady, non-Darcy flow.
Keywords: Non-Darcy flow chaos power spectrum correlation dimension Lyapunov indices
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