有限时间Lyapunov指数的高精度计算新方法

摘要

Aiming at the shortcomings of current method of calculating finite-time Lyapunov exponent (FTLE), such as low accuracy, inability to obtain boundary values, etc., a method of highly accurately computing FTLE is proposed based on dual number theory. Firstly, the weakness and disadvantages of the finite difference method used widely for computing FTLE are described. Secondly, the dual number theory is introduced to evaluate the derivatives accurately and efficiently, and its distinct virtues are also presented. The computation of Cauchy-Green deformation tensors for a dynamical system is transformed into a numerical integration problem of solving the nonlinear ordinary differential equation in dual number space by the new method. Finally, the proposed method is applied to typical pendulum system and nonlinear Duffing oscillator separately. The results of simulation experiments indicate that the new method is effective, convenient and accurate for computing the field of FTLE, from which Lagrangian coherent structures can be identified successfully.%针对目前有限时间Lyapunov指数(FTLE)计算方法准确度不高和无法获得边界值的问题，基于对偶数理论提出了一种新的高精度计算方法。首先描述了基于有限空间差分方法计算FTLE的缺点和问题；其次介绍了基于对偶数理论的高精度导数计算方法及其显著优点，并将动力系统的柯西-格林形变张量计算问题转化为对偶数空间中非线性微分方程数值求解问题；最后对单摆和非线性Duffing振子两个典型物理动力系统进行了数值实验。结果表明：基于对偶数理论的新方法能有效、方便和高精度地计算出有限时间Lyapunov指数场，并成功识别出所包含的拉格朗日相关结构。