Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

作     者：Julio Cesar Bastos de Figueiredo Luis Diambra Coraci Pereira Malta 

作者机构：不详 

出 版 物：《Applied Mathematics》 (应用数学（英文）)

年 卷 期：2011年第2卷第4期

页      面：436-443页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Chaotic Attractor Statistical Measure Numerical Integration 

摘      要：Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the difference between the time series for steps h and h/2 becomes smaller as h decreases. Unfortunately, this convergence criterium is unsuitable in the case of a chaotic solution, due to the extreme sensitivity to initial conditions that is characteristic of this kind of solution. We present here a criterium of convergence that involves the comparison of the attractors associated to the time series for integration time steps h and h/2. We show that the probability that the chaotic attractors associated to these time series are the same increases monotonically as the integration step h is decreased. The comparison of attractors is made using 1) the method of correlation integral, and 2) the method of statistical distance of probability distributions.