Date Published: January 31, 2017
Publisher: Public Library of Science
Author(s): ChaoJiu Da, Fang Li, BingLu Shen, PengCheng Yan, Jian Song, DeShan Ma, Zhen Jin.
We conducted an exploratory study of the detection of a sudden change of the field time series based on the numerical solution of the Lorenz system. First, the time when the Lorenz path jumped between the regions on the left and right of the equilibrium point of the Lorenz system was quantitatively marked and the sudden change time of the Lorenz system was obtained. Second, the numerical solution of the Lorenz system was regarded as a vector; thus, this solution could be considered as a vector time series. We transformed the vector time series into a time series using the vector inner product, considering the geometric and topological features of the Lorenz system path. Third, the sudden change of the resulting time series was detected using the sliding t-test method. Comparing the test results with the quantitatively marked time indicated that the method could detect every sudden change of the Lorenz path, thus the method is effective. Finally, we used the method to detect the sudden change of the pressure field time series and temperature field time series, and obtained good results for both series, which indicates that the method can apply to high-dimension vector time series. Mathematically, there is no essential difference between the field time series and vector time series; thus, we provide a new method for the detection of the sudden change of the field time series.
In 1953, Hadamard solved the Cauchy problem of Laplace’s equation by formulating the instability of the solution of the differential equation for the first time, and constructed the counterexample, which shows that the differential equation is sensitive to the initial value. In the middle of the20th century, Thom studied the singularity theory of a mapping on the differentiable manifold and classified the singularities of the function in Euclidian space to obtain a series of conclusions, which include the famous transversality theorem. These conclusions form the mathematical foundation of sudden change theory. Zeeman conducted a systematic study of sudden change theory and expanded the theory’s application. In 1963, Lorenz analyzed the nonlinear effect in the convection motion equation of the atmosphere and found that when the parameters of the equations are particular values, the path motion becomes complex and uncertain, which leads to the unpredictability of the path. The study’s significance has two aspects: First, the study confirmed the counterexample of Hadamard from the viewpoint of a numerical experiment. Second, tests conducted during the study illustrated that the climate demonstrates the phenomenon of sudden change.
We used the proposed method to detect a sudden change of a meteorological field time series, which is the original space-time field time series. We constructed the time series and detected a sudden change. Methods used to study the meteorological field are typically various indices that reduce the original space-time field dimensions.
Using the numerical solution of the Lorenz system, the surfacedemarcating the regions to the left and right of the equilibrium point of the Lorenz system, and some mathematical techniques, we proposed a new method to detect a sudden change of the field time series. The method has good practicability and universality, but requires certain mathematical and physical techniques that are based on an adequate understanding the geometric characteristics and physical properties of the field time series. Whether using the test of the Lorenz system or the test of the practical meteorological field time series, a sudden change can be detected. Thus, our proposal is innovative, both in the method and theory.