用线性化方程研究混沌系统可预报性的讨论

Investigation of Studying Predictability of Chaotic System by Using its Linearized Equations

采用著名的Lorenz方程组讨论了用线性化方程组研究混沌系统可预报性的局限性。结果表明 ,Lorenz方程组的线性化方程组刻划了小扰动的线性发展 ;而Lorenz方程组本身只在有限时段内 ,刻划初始小扰动的线性发展 ,随着时间的发展 ,Lorenz系统开始出现混沌 ,描述了初始扰动的非线性发展 ,此时其线性化方程组失效。另外 ,通过比较小扰动和大扰动导致系统进入混沌的时间表明 ,扰动越小 ,时间跨度越小 。

The well-known Lorenz equations are used to study the limitation that the predictability of chaotic system has been inˉvestigated by using the linearized equations.It is demonstrated that the linearized version of Lorenz equations describe the evolutions of sufficiently small perturbations,while Lorenz equations themselves characterize the development of the small perturbations only during the finite short time interval.With the development of time,Lorenz system begins to get into chaotic state,which characterizes the nonlinear evolution of the initial perturbation.In this case,the linearized equations do not hold.Besides,comparisons between the times when the evolutions of the small and large perturbations get into chaotic state show that the small initial perturbations and the short time intervals support the holding of the linear approximation to the chaotic Lorenz system.