Lorenz-96模型中的泛混沌
Extensive Chaos in the Lorenz-96 Model
摘要：
通过计算Lorenz-96模型的分形维数随系统参数的变化，研究了该模型的高维混沌动力学行为. Lorenz-96模型是一种时间连续、空间离散的模型，由Edward Lorenz首先提出，用于研究空间扩展混沌系统（如大气）的预测。首先，我们通过增加系统的大小，同时保持外部强迫的大小不变来探索时空混沌极限。其次，我们通过增加外部强迫，同时保持系统大小固定，探索强驱动极限。由于系统的大小增加了小值的强迫，我们发现动态状态，交替周期和混沌动力学。混沌的窗口是广泛的，平均而言，与外延的相对偏差在20%的顺序。对于中间值的强迫，我们发现混沌动力学的所有系统大小过去的临界值。分形维数表现出最大的偏离外延的顺序为5%的系统大小的小变化，并随着系统大小的增加而非单调下降。描述的长度尺度从外延和自然的混沌长度尺度的偏差是近似相等的支持的建议，从外延的偏差是由于增加混沌自由度的系统大小的增加。随着强迫的增加，在恒定的系统尺寸的分形维数表现出幂律依赖。幂律的行为是独立的系统的大小和量化的混沌自由度的大小随着强迫的增加，我们比较与空间特征的模式。
英文摘要：
We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Edward Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and decreases non-monotonically with increasing system size. The length scale describing the deviations from extensivity and the natural chaotic length scale are approximately equal in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. As the forcing is increased at constant system size the fractal dimension exhibits a power-law dependence. The power-law behavior is independent of the system size and quantifies the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.
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